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Abstract
Enhanced photonic microwave generation by using a filtered optical feedback in an optically injected semiconductor laser operating at period-one (P1) dynamics is numerically demonstrated. In the simulation, the frequency tunability of the generated narrow-linewidth photonic microwave with the filtered optical feedback has been investigated. The results show that the frequency of the narrow-linewidth photonic microwave can be widely tuned by adjusting the injection parameters only or adjusting both the injection parameters and the center frequency of the filter. Moreover, the influence of the delay time, feedback strength, filter bandwidth and detuning on the linewidth, side-peak suppression and phase noise of the generated microwave have also been investigated in detail. The results show that with increasing feedback strength or delay time, evident reduction of the linewidth is observed. The side-peak suppression also increases with increasing feedback strength; however, side-peak suppression decreases with increasing feedback delay time. In addition, the linewidth reduction and side-peak suppression are relatively robust to the filter detuning, especially for higher feedback strengths and microwave frequencies. This is mainly attributed to the self-adaptive shifting of the red-shifted cavity resonance frequency to the center frequency of the filter in the FOF configuration.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Microwave photonics technologies consisting of photonic microwave generation, processing, control and distribution, have drawn considerable attention due to their potential applications in the fields of optical wireless communication, sensors and radar [1–4]. From the perspective of photonic microwave generation, it provides various promising advantages over their electronic counterparts, such as capability of generating millimeter-wave, broad and continuous frequency tunability, and long-distance microwave distribution over the fiber-link [5,6]. Many techniques for photonic microwave generation have been reported up to now, which include direct modulation [7,8], optical heterodyne oscillators [9,10], external modulation [11,12], mode-locked semiconductor lasers [13,14] and period one (P1) dynamics.
P1 dynamic in optically injected semiconductor lasers have attracted significant research interests for photonic microwave generation [15–23]. Under the scenario of continuous-wave injection, the laser can operate under many different nonlinear dynamics, such as injection-locking, P1, period-doubling, quasi-periodic, and chaos. P1 dynamic is obtained when a stably locked semiconductor laser experiences a Hopf-bifurcation [24], and it has been investigated numerically and experimentally in distributed feedback lasers [25,26], quantum-dot lasers [27,28] and vertical-cavity surface-emitting lasers (VCSELs) [20,29]. The main feature of P1 dynamic in an optical spectrum is that the optically injected semiconductor laser has an equally spaced set of frequencies, of which the two most dominant frequencies are the regenerative frequency from the optical injection and the red-shifted cavity resonance frequency (RSCRF). The beating of these frequencies would give birth to a microwave signal.
The photonic microwave generated based on P1 dynamic (also known as the P1 microwave) has several advantages. For instance, P1 microwave generation holds a low cost due to the all-optical components’ configuration [30,31], a widely tunable range from a few gigahertz to tens or even hundreds of gigahertz [27,32], and an enhanced power efficiency because of a nearly single sideband spectrum [33]. However, P1 dynamic inherently contain phase noise due to spontaneous emission noise in the semiconductor lasers. It would cause adverse effects such as reductions of the signal-to-noise ratio and broadening the microwave linewidth [34]. To overcome these drawbacks, some schemes for linewidth reduction have been proposed. As reported in [15], a double-locked laser diode can generate microwaves at frequencies ranging from 9.5GHz to 17.1 GHz with a linewidth below 1 kHz. Tunable P1 microwaves with frequency from 10GHz to 23GHz and linewidth at a range of 40 kHz to 120 kHz have been experimentally obtained by employing an optoelectronic feedback loop [16]. Feeding a portion of the laser output back into the laser cavity from an external mirror can narrow the linewidth of the laser [35,36]. Another benefit for using optical feedback as the all-optical approach in linewidth reduction is that it does not hold the frequency limitations of optoelectronic feedback. Therefore, optical feedback is regarded as one of the most potential ways for linewidth reduction. It is worth mentioning that conventional single optical feedback (SOF) can causes multiple side peaks (SPs) in the power spectra due to external cavity modes (ECMs), which can induce purity degeneration and therefore is undesired. Recently, lots of advanced schemes based on optical feedback are proposed [34,37–39]. In [37], with the polarization-rotated feedback, the microwave linewidth is reduced to ~15 kHz with SP suppressed of 15–20 dB. With the help of an additional optoelectronic feedback, a narrower linewidth of ~3 kHz is obtained at a microwave frequency of 6 GHz. P1 microwave with SP suppression and a linewidth less than 50 kHz has been experimentally demonstrated in an optically injected semiconductor laser with a double optical feedback (DOF) [38]. Theoretical investigation of linewidth reduction and SP suppression using conventional optical feedback (SOF and DOF) has also been reported [34]. Recently, we also experimentally demonstrated [39] that the conventional optical feedback can improve the stability of the P1 microwave and reduce the linewidth in an optically injected VCSEL. In addition, the linewidth of the microwave decreases with increasing feedback strength until an optimal feedback strength in SOF and DOF configurations is reached [34,39]. Further increasing feedback strength, P1 dynamic is gradually collapsed with more complex dynamics. We recently proposed the linewidth reduction of P1 microwaves using a filtered optical feedback (FOF) [40]. The results show that the FOF is outperformed the conventional optical feedback in terms of linewidth reduction and SP suppression because the narrow bandpass filter in the feedback loop can effectively control the ECMs under the profile of the filter passband [41].
In this paper, we extend our earlier work reported in [40]. P1 microwave generation in an optically injected semiconductor laser with FOF is systemically investigated and analyzed, including the frequency tunability, effect of feedback strength, delay time, filter bandwidth and filter detuning on linewidth reduction, SP suppression and phase noise. Following this introduction, the simulation model is presented in section 2. Detailed numerical results and discussions are reported in section 3. Finally, the conclusions are drawn in section 4.
2. Model
Figure 1 shows the scheme for P1 microwave generation in a semiconductor laser subject to optical injection and FOF. Continuous-wave optical signal generated by a master laser (ML) is unidirectionally injected into a slave laser (SL). A part of the SL output is fed back into its cavity through a feedback loop. Different from the conventional optical feedback, the feedback loop in the FOF configuration includes a narrow bandwidth optical filter. The optical filter is modeled by a Lorentzian filter, the dynamics of the SL with terms of optical injection and filtered feedback is modeled by normalized single-mode rate equations [34,40–43],
where, a, n and F denote the normalized field amplitude, carrier density and normalized field amplitude of the filter’s output respectively. γc is the cavity decay rate, γs is the spontaneous carrier relaxation rate, γn is the differential carrier relaxation rate, γp is the nonlinear carrier relaxation rate, b is the linewidth enhancement factor, and J is the normalized bias current. ξ represents the optical injection strength. fi and υ are the offset frequency of optical injection and the center frequency of filter with respect to the free-running frequency of the SL f0, respectively. υ is also named filter detuning. Λ is the half-width at half-maximum (HWHM) of the filter and is used to quantify the filter bandwidth. η and τ are the feedback strength and round-trip delay time, respectively. The typical parameters values used in [34] are adopted in this simulation, where γc = 5.36 × 1011 s−1, γs = 5.96 × 109 s−1, γn = 7.53 × 109 s−1, γp = 1.91 × 1010 s−1, b = 3.2, J = 1.222 and f = 193.41 THz. With these parameters, the relaxation oscillation frequency of the free-running laser is given by fr = (2π)−1(γcγn + γsγp)1/2 ≈10.25 GHz.
Fig. 1 Schematic diagram for P1 microwave generation in a semiconductor laser with optical injection and FOF. OI: optical isolator, OC: optical coupler, Cir: circulator, BF: band-pass filter, PC: polarization controller, Att: attenuator, ML: master laser, SL: slave laser, PD: photodetector.
The spontaneous emission noise is modelled by a Langevin fluctuating force χ, as illustrated in Eqs. (4)-(6), of which the real and imaginary parts are mutually independent. Rsp is used to describe the strength of χ [44],
Here, Rsp is set at 5.99 × 1019 V2m−1s−1 to approach the linewidth of the generated microwave observed in the experiment [16]. In this simulation, the filter bandwidth Λ is fixed at 160 MHz and the feedback delay time τ of the FOF configuration is set to 2.4 ns, unless stated otherwise. For comparison, the configurations with conventional optical feedback are also numerically studied. To model conventional optical feedback, Eq. (3) does not need to be included and the feedback term F in Eq. (1) can be written aswhere τ1 and τ2 are the feedback delay time from cavity 1 and 2 for DOF, respectively. In our simulation, a second-order Runge-Kutta with a time step of 1 ps is used to numerically solve the rate Eqs, the microwave linewidth is investigated with a time span of 1 ms, while the other is investigated with a time span of 5 μs for the sake of simplicity.3. Results and analyses
3.1 The tunability of the P1 microwave
In the FOF configuration, the filter passband should always cover the RSCRF to ensure that the component of RSCRF is fed back into the SL. Because the generated P1 microwave is mainly attribute to the beating of the RSCRF and regenerative frequency and the spontaneous emission noise in the SL mostly affect the linewidth of the RSCRF, rather than the regenerative signal. Two methods can be used to tune the microwave frequency: one method is to tune the filter center frequency in line with the varying RSCRF; the second method is to fix the RSCRF, but to change the injection parameters.
In the former method, the injection parameters can be selected freely to achieve a wide microwave frequency tunability, while the central frequency of filter should be carefully adjusted to match the varying RSCRF. Figure 2 illustrates the optical spectra and RF spectra of the SL with different RSCRF under the configurations of injection only and both injection and FOF. Here, the injection parameters are selected according to the maximum microwave power curve reported in [34], and the feedback strength is fixed at η = 0.012 in the FOF configuration. The optical and RF spectra are calculated by adopting the fast Fourier transform (FFT) to a(t) and |a(t)|2, respectively. The linewidth is calculated from the smoothed RF spectrum by the default average moving in MATLAB. Figure 2(a) shows the scenario with (fi, ξ) = (7 GHz, 0.12). When the SL is subject to injection only, the optical spectrum contains a set of equally spaced frequencies, as shown in Fig. 2(a1). The two dominant components at fi and fRS (RSRCF) are the regenerative injection signal and the red-shifted cavity resonance signal, respectively. The laser operates at P1 dynamic. The beating of these frequencies generates photonic microwave with a P1 microwave frequency of 18.08 GHz (fi - fRS). As shown in the RF spectrum in Fig. 2(a2), a relatively wide linewidth of 13.2 MHz is observed due to the spontaneous emission noise. When the FOF with υ = −11GHz is deployed in the optically injected SL, the P1 microwave frequency shifts slightly to 18.17 GHz and the linewidth is reduced from 13.2 MHz to 134.1 kHz [see Figs. 2(a3) and 2(a4)]. In Fig. 2(b) with (fi, ξ) = (15 GHz, 0.17), the generated microwave frequency in the injection only scenario is increased to 24.06 GHz (the RSCRF is −9.06 GHz), and a linewidth of 22.4 MHz is acquired [See Figs. 2 (b1) and (b2)]. For the scenario of FOF with the same injection parameters, the filter detuning is shifted to −9 GHz close to the RSCRF. As shown in Figs. 2(b3) and 2(b4), a P1 microwave with a frequency of 24.13GHz and a narrower linewidth of 34.4 kHz is observed. In Fig. 2(c) with (fi, ξ) = (55.5 GHz, 0.48), in the injection only scenario, a microwave frequency as high as 59.78 GHz (the RSCRF is −4.28 GHz) is obtained with a linewidth of 85 MHz [See Fig. 2(c1) and (c2)]. In its counterpart with FOF, by setting υ = −4.5GHz, a P1 microwave with a frequency of 59.7 GHz and a linewidth of 22.6 kHz is achieved [See Figs. 2(c3) and 2(c4)]. It is noted that the linewidth of the microwave is dependent on the injection parameters. The detailed relationship between the linewidth and the injection parameters in optical injection only configuration can be found in [34]. It is also worth mentioning that the SP is also well suppressed in the FOF configuration. Moreover, the separation of the SP to the fundamental frequency in Figs. 2(a4-c4) are 298 MHz, 320 MHz and 300MHz respectively, and all of them are less than reciprocal of the feedback delay time (1/τ) which is experimentally observed in the SOF configuration [33]. This is probably due to the filter effect.
Fig. 2 Optical spectra and RF spectra of the SL operating at P1 dynamic subject to injection only (the first and second columns), and injection and FOF (the third and fourth columns). The injection parameters (fi, ξ) = (a) (7 GHz, 0.12), (b) (15 GHz, 0.17), (c) (55.5 GHz, 0.48). The filter detuning in the FOF is set as υ = (a) −11 GHz, (b) −9 GHz, (c) −4.5GHz. η = 0.012. The RSCRF fRS is labeled in the optical spectra.
The RSCRF is mainly dependent on the injection parameters, and when the RSCRF is fixed during the adjusting of injection parameters, tunability of the microwave frequency can also be achieved without adjusting the filter detuning. Under this scenario, the microwave frequency is linearly changed according to the injection frequency detuning. To help select the injection parameters, the map of the RSCRF and the fundamental frequency of P1 microwave in the SL with injection only as a function of the injection strength and frequency detuning is plotted in Fig. 3. Different from the frequency of the regenerative signal, the RSCRF witnesses a complex variation with the change of injection strength and frequency detuning. As shown in the green areas in Fig. 3, when the injection strength is small, the unusually positive RSCRF can be observed due to the injection pulling effect [33]. The RSCRF is generally negative and its absolute value increases with the increasing of injection strength but decreases with the increasing of frequency detuning. It is worth mentioning that P1 microwave frequencies with a fixed RSCRF can still hold a wide range of tunability by appropriately selecting the injection parameters. For instance, the curve of RSCRF = −9 GHZ covers a wide range of microwave frequencies from less than 16 GHz to more than 48 GHz. In other words, the tunability of the P1 microwave in the FOF configuration with a fixed center frequency of an optical filter can be achieved by appropriately adjusting the injection parameters. Figure 4 illustrates the optical spectra and RF spectra of the P1 microwave with a fixed RSCRF but different microwave frequencies in the FOF configuration. The injection parameters are selected properly according to the results in Fig. 3 (the RSCRF is around −9 GHz). The parameters of FOF are fixed as (υ, Λ, η, τ) = (−9 GHz, 160 MHz, 0.020, 2.4 ns). In Fig. 4(a) with (fi, ξ) = (7.5 GHz, 0.10), the SL exhibits P1 dynamic with a RSCRF of −9.11 GHz, which corresponds to a microwave frequency of 16.61 GHz. A narrow linewidth of ~37 kHz is observed. In Fig. 4(b) with (fi, ξ) = (23 GHz, 0.265), the microwave frequency increases to 32.13 GHz (corresponding to a RSCRF −9.13 GHz) and a linewidth of 24.2 kHz is observed, as shown in Fig. 4(b2). Further changing the injection parameters to (fi, ξ) = (31GHz, 0.365), a P1 microwave with a frequency of 40.13GHz and narrow linewidth of 27.1 kHz is generated. Moreover, although the actual RSCRF in the three scenarios is not well matched with the fixed filter center frequency, a relatively narrow linewidth can still be obtained. The results indicate that the small mismatching of RSCRF to the central frequency of the filter will not affect the linewidth reduction significantly. Detailed investigation of the filter detuning on the performance of P1 microwave will be presented later.
Fig. 3 RSCRF and microwave frequency (dashed curve) of the SL with injection only as a function of the injection strength and frequency detuning.
Fig. 4 Optical spectra and RF spectra of the SL operating at P1 dynamic with optical injection and FOF. (a) (fi, ξ) = (7.5 GHz, 0.1), (b) (23GHz, 0.265), and. (b) (31GHz, 0.365). The parameters of FOF are set as (υ, Λ, η, τ) = (−9 GHz, 160 MHz, 0.020, 2.4 ns). The RSCRF fRS is labeled at the optical spectra.
3.2 Performance of P1 microwave versus feedback strength
Effect of the feedback strength on the generated P1 microwave in the FOF configuration is also investigated. It has been reported that feedback strength plays a significant role in the linewidth reduction of P1 microwaves under conventional optical feedback [34,38,39]. In our previous work [40], we have demonstrated that the characteristics of P1 microwaves can be further optimized in the FOF configuration by employing stronger feedback strength than conventional optical feedback configurations. To get an insight into the stability of P1 dynamic under FOF, the bifurcation diagram of the SL as a function of the feedback strength is calculated and displayed in Fig. 5, together with the conventional ones (SOF and DOF). The injection parameters in Fig. 5 are the same as those used in Fig. 4(b). As shown in Fig. 5(a) with the FOF configuration, the P1 dynamic can be retained in the whole range of feedback strength (0 < η < 0.2) since the bandpass filter has attenuated the effect of the ECMs on the dynamics of the SL. Meanwhile, for the SOF and DOF configurations, the SL can only remain at P1 dynamic with feedback strength η < 0.035 [See Figs. 5 (b) and (c)]. When η >0.035, the SL is driven into more complexity dynamics, such as period-doubling, quasi-periodicity and chaos.
Fig. 5 Bifurcations of the optically injected SL subject to (a) FOF, (b) SOF and (c) DOF without the noise. The parameters (fi, ξ, τ, τ1, τ2) are set as (23 GHz, 0.265, 2.4 ns, 2.4 ns, 3 ns).
Figure 6 illustrates the RF spectra of the optically injected SL subject to FOF with the different feedback strengths. Here, the same injection parameters as Fig. 4(b) are used. In Fig. 6(a) with η = 0.005, the linewidth of the P1 microwave has been reduced from 34 MHz (without any feedback) to 379 kHz. Two small SPs equally separated from the fundamental frequency are also observed. To quantify the SP in the RF spectrum, the side-peak suppression coefficient (SPSC), defined as the ratio of the power of the fundamental microwave frequency to the maximum SP, is calculated. The simulation results show that the value of the SPSC changes with different simulation time spans (not shown here). The longer the time span, the higher the SPSC obtained. In Fig. 6, a time span of 1 ms is used, and the SPSC in Fig. 6(a) is 23.4 dB. When the feedback strength is increased to η = 0.020, the linewidth is reduced to 24.2 kHz and the SPSC increases to 30.4 dB [See Fig. 6(b)]. Further increasing the feedback strength to η = 0.035, as shown in Fig. 6(c), the linewidth and SPSC are 7.3 kHz and 33.8 dB, respectively. Finally, in Fig. 6(d) with η = 0.05, a high-quality P1 microwave with linewidth 5.9 kHz and SPSC 38.1 dB is achieved. Evidently, relatively strong feedback strengths can benefit the P1 microwave in both linewidth and SPSC. In terms of SP suppression, ultrashort optical feedback can also be used for side peak reduction because side peaks are very far away from the period-one microwave oscillation [45]. The use of ultrashort optical feedback can be applied in photon integration. However, it is difficult to obtain extra-short cavity feedback in a conventional configuration. In addition, the SPs suppression in extra-short cavity feedback is very sensitive to the feedback phase, which may compromise the stability. While the SPs suppression is not sensitive to the feedback phase in filter optical feedback (not shown here).
Fig. 6 RF spectra of the SL with FOF. The feedback strength η = (a) 0.005; (b) 0.020; (c) 0.035 and (d) 0.050. The insets are the enlarged of the fundamental frequency with a frequency span of 6 MHz.
Figure 7 presents a detailed investigation about the influences of feedback strength on the characteristic of P1 microwaves in the FOF scenario. All three sets of injection parameters are selected in such way that the RSCRF is fixed around −9GHz. The parameters of the filter are fixed at (Λ, υ) = (160 MHz, −9 GHz). Figure 7(a1) shows the linewidth of the microwave as a function of the feedback strength. The injection parameters (fi, ξ) are (7.5 GHz, 0.1) and the generated microwave frequency is around 16.5 GHz. The result shows that with increasing feedback strength, the linewidth gradually reduces to an optimal linewidth at η = 0.025 before rising again. This is because P1 dynamic is destroyed gradually and when η is over 0.032, the SL starts to work in the period-double regime. With respect to the maximum allowed feedback strength of η = 0.012 and η = 0.016 for P1 dynamic in the SOF and DOF configuration respectively (not shown here), the maximum allowed feedback strength in FOF configuration has been significantly enhanced. As for the SPSC, it presents a clear increasing trend with increasing feedback strength until it reaches the maximum value at η = 0.025 before decreasing as the feedback strength increases further. Figures 7 (b1) and (b2) shows the linewidth and SPSC evolutions when the injection parameters (fi, ξ) are (15 GHz, 0.17) (corresponding to a microwave frequency of ~24 GHz) [see Fig. 2(b)]. It shows that the linewidth decreases with increasing feedback strength until the feedback strength reaches around 0.038. Further increasing the feedback strength has little effect on linewidth reduction until the P1 is destroyed. The SPSC also presents evident climbing with increasing feedback strength. In Fig. 7(c) with (fi, ξ) = (31 GHz, 0.365), the linewidth and SPSC of the P1 microwave with frequency ~40 GHz is investigated. The results show that the linewidth and SPSC present a same trend as those in Figs. 7(b1) and 7(b2). The fluctuations in the linewidth and SPSC evolution in Fig. 7 are also observed, which is due to high noise level used in the simulation.
Fig. 7 Linewidth (the first row) and SPSC (the second row) as a function of the feedback strength. The (fi, ξ) are set as (a) (7.5GHz, 0.1), (b) (15 GHz, 0.17), (c) (31 GHz, 0.365).
The tolerance of P1 dynamic to the feedback strength as a function of the injection strength with conventional optical feedback (COF) and FOF is illustrated in Fig. 8. Here, fi is fixed at 15GHz and the filter detuning is adjusted according to the variation of RSCRF related to the injection strength. Red area presents that the SL operates at complex dynamics (such as, period doubling or chaos), which indicates that P1 dynamic can be destroyed and enter complex dynamics by higher feedback strength. Green area marks that the SL remains at P1 dynamics with COF or FOF. In blue area, the SL will be in complex dynamics if it is subject to COF, but the SL stay at P1 dynamics with FOF. The result is consistent with that in Fig. 5 that P1 dynamic can tolerate higher FOF. The map also shows that the robustness of P1 dynamic to optical feedback is improved (P1 dynamic can withstand stronger optical feedback) with increasing injection strength. It is worth mentioning that the SL operates at an injection-locking state when the injection strength is over the upper boundary [See Fig. 3].
Fig. 8 Map of the dynamics of the SL versus the injection strength and feedback strength. fi = 15 GHz and τ = 5 ns.
To characterize the extent of microwave stabilization, phase noise is also investigated. The phase noise can be quantified by the phase noise variance which is estimated by integrating the averaged single sideband of the power spectrum at the microwave frequency [34,45,46]. Figure 9 plots the phase variance as a function of the feedback strength with SOF and FOF. The injection parameters (fi, ξ) are (15 GHz, 0.17) and τ = 2.4 ns. For comparison, linewidth and SPSC verse the feedback strength are also presented in inset (a) and (b), respectively. The results show that with the increase of the feedback strength, the phase variance in the FOF configuration witnesses an evident decreasing and reach the minimum value at η = 0.065. Furtherly increasing η, the phase variance increases again, which can be contributed to the SPSC decreasing observed in the inset(b). While the phase variance in the SOF configuration (shown as the circles), it firstly decreases slowly with the increase of η, but it increases when η > 0.015. It is obvious that the FOF configuration holds a much smaller phase variance than the conventional single optical feedback (SOF) for higher feedback strength. The insert (a) shows that the linewidth in the FOF and SOF configuration witness an evidently reduction with the increasing of the feedback strength when η < 0.03. It is worth noted that even the linewidths are comparative for both SOF and FOF, the phase noise variance in the FOF is still much smaller than that in the SOF. This is because the SP suppression in the FOF configuration is higher than that in the SOF configuration [See the insert (b)]. For higher η, the linewidth in the SOF configuration increases with the increase of η and the SL is gradually driven out of the P1 dynamics, however the linewidth is continuously decreased with the increase of η in the FOF configuration.
Fig. 9 Phase variance versus the feedback strength. (fi, ξ) = (15 GHz, 0.17), τ = 2.4 ns. Inset (a) linewidth verse the feedback strength; inset (b) SPSC verse the feedback strength.
3.3 Performance of P1 microwave versus the filter bandwidth
Figure 10 illustrates the influence of the filter bandwidth on the linewidth and SPSC of the generated microwave. In Fig. 10(a) with (ξ, fi,) = (0.10, 7.5 GHz) and η = 0.025, for small filter widths, the linewidth decreases rapidly as the filter width increases until the filter width reaches 0.15 GHz, and then the linewidth remains at a minimum in the range of 0.15 GHz to 0.25 GHz. This is because when the filter bandwidth is small, increasing feedback power dominates linewidth reduction. The feedback power increases with increasing filter bandwidth, therefore the linewidth decreases with increasing filter bandwidth until it reaches its minimum value. At this filter bandwidth, all the power of the RSCRF passes through the filter, further increasing the filter bandwidth does not substantially increase the feedback power until the power of the external cavity modes also passes through the filter. When Λ > 0.25 GHz, the linewidth increases with further increasing filter bandwidth. This is because the SL is gradually driven out of P1 dynamic as more ECMs feedback into the SL and the maximum feedback strength allowed for the P1 dynamic is degraded. In Figs. 10(b) with (ξ, fi) = (0.17, 15 GHz) and η = 0.035 and 10(c) with (ξ, fi) = (0.365, 31 GHz) and η = 0.04, the linewidth first drops rapidly and then remains almost constant with increasing filter bandwidth since the feedback strength is not over the maximum feedback strength that P1 dynamic can tolerate. For SPs suppression, significant reductions of the SPSC with increasing filter bandwidth are observed in all three cases due to the enhancement in the ECMs [See Figs. 10(a2)-10(c2)]. The results show that the narrower filter bandwidth can benefit the performance of the microwave. An etalon can be used as such a narrow bandwidth filter, which has been experimentally demonstrated [41]. The simulation results also indicate that the filtered feedback with 2 GHz bandwidth filter can also present a better performance than that conventional feedback in terms of linewidth reduction and SP suppression (no shown here).
Fig. 10 Linewidth (the first row) and SPSC (the second row) of the photonic microwave generated in the optically injected SL subject to FOF as a function of the filter width. The parameters (ξ, fi, η) are set as (a) (0.1, 7.5GHz, 0.025), (b) (0.17, 15 GHz, 0.035), (c) (0.365, 31 GHz, 0.04).
To give an insight into the influences of filter bandwidth on the maximum feedback strength that P1 dynamic can tolerate, the dynamics evolution versus the feedback strength and filter bandwidth with the injection parameters (ξ, fi) of (0.17, 15 GHz) is calculated and plotted in Fig. 11. The results show that when the filter bandwidth is relatively wide (> 14GHz), P1 dynamic in the FOF configuration can only endure a feedback strength of less than 0.035, which is similar to the SOF scenario. With decreasing filter bandwidth, P1 dynamic can withstand stronger feedback strengths. This is because the ECMs under the profile of the filter passband is related to the filter bandwidth, and the enhancement of the ECMs makes P1 dynamic easier to be destroyed. A small island where the SL remains at P1 dynamic with high feedback strength and wide filter bandwidth also appears in Fig. 11, which is not fully understood and needs further investigation.
Fig. 11 Dynamics of the SL with FOF as a function of the feedback strength and filter width. The parameters (ξ, fi) are set to (0.17, 15 GHz).
Figure 12 shows the phase variance as a function of the filter bandwidth. With the expansion of the filter bandwidth, the phase variance experiences a sharp decreasing at the beginning and achieve the minimum value at the filter bandwidth of ~250 MHz, which is in line with the significantly reduction of the linewidth observed in Fig. 10(b1). Further increasing the filter bandwidth, the phase variance starts to increase. This is because the evident decreasing of the SPSC [See Fig. 10(b2)].
Fig. 12 Phase variance versus the filter bandwidth. (fi, ξ) = (15 GHz, 0.17), τ = 2.4 ns and η = 0.035.
3.4 Performance of P1 microwave versus the filter detuning
The influences of filter detuning on the generated microwave is also inspected and the results are presented in Fig. 13. The top row of Fig. 13 illustrates the linewidth of the P1 microwave as a function of filter detuning. In Fig. 13(a1) with (ξ, fi) = (0.1, 7.5GHz) and η = 0.025, the frequency of the P1 microwave is around 16.5 GHz, and evident linewidth reduction emerges when the filter detuning is set from −10 GHz to −8.75GHz (the width is ~1GHz). When the filter detuning is out of this range, wider linewidth (even over 1 MHz) is observed. In Fig. 13(b1) with (ξ, fi) = (0.17, 15 GHz) and η = 0.035, the P1 microwave with a frequency around 24GHz has narrower linewidth when the filter detuning is set from −10.75 GHz to −7.75 GHz (the width is ~3 GHz). Finally, in Fig. 13(c1) with (ξ, fi) = (0.365, 31 GHz) and η = 0.04, a narrow-linewidth microwave with a frequency of around 40 GHz is acquired when the filter detuning is set from −12.75 GHz to −7.25 GHz (the width is ~5.5 GHz). It is obvious that a significant mismatch between the center frequency of the filter to the initial RSCRF (−9 GHz) calculated from the injection only configuration is observed. This is because the RSCRF opts to stabilize at its ECMs [45]. The power of the adjacent ECM is enhanced when the filter center frequency is close to the adjacent ECMs, which would motivate the RSRCF shift itself to the adjacent ECM. Therefore, the RSCRF can self-adapt to some extent by shifting itself to around the center frequency of the filter [See the second row of Fig. 13, the red lines indicate the center frequency of the filter]. We name this phenomenon as self-adaptive frequency shift. Because the RSCRF self-adaptive frequency shift, the generated microwave frequency also shifts with varying the filter detuning, as shown at the third row of Fig. 13. The microwave frequencies display a step (related to the 1/τ) increase with increasing absolute filter detuning. The bottom row of Fig. 13 displays the SPSC as a function of the filter detuning. The results demonstrate that good suppression of the SPs and linewidth reduction almost hold the same range of filter detuning except for some bad points of lower SPSC emerged at the boundary of the range. The bad points correspond to the unstable points where the RSCRF is about to transfer to the other ECM, which will be explained in more detail in Fig. 14.
Fig. 13 Linewidth (top row), RSCRF (second row), microwave frequency (third row) and SPSC (bottom row) as a function of the filter detuning. The parameters (ξ, fi, η) are set as (a) (0.1, 7.5GHz, 0.025), (b) (0.17, 15 GHz, 0.035), (c) (0.365, 31 GHz, 0.04).
Fig. 14 Maps of (a) microwave frequency and (b) SPSC as a function of the feedback strength and filter detuning, (ξ, fi) = (0.17, 15 GHz).
Figures 14(a) and 14(b) display the maps of the microwave frequency and SPSC as a function of the feedback strength and filter detuning, respectively. The injection parameters (ξ, fi) are (0.17, 15 GHz) and the generated microwave frequency is around 24 GHz. As shown in Fig. 14 (a), the change of the microwave frequency is mainly due to the filter detuning. In the self-adaptive frequency shift regime, the range of the frequency shift increases with increasing the feedback strength. The map also clearly shows that the microwave frequency is a step-like change when the center frequency of the filter is detuned. This is because the RSCRF is only stable at an ECM near the center frequency of filter. The results in Fig. 14(b) show that the regime with the higher SPSC is almost overlapped with that with the self-adaptive frequency shift. However, in the self-adaptive frequency shift regime, the lower SPSC (bad points at the bottom row of Fig. 13) happens periodically with increasing the filter detuning at a fixed feedback strength. The lower SPSC occurs at the filter detuning where the microwave frequency changes abruptly,which clearly indicates that the lower SPSC takes place when the RSCRF is about to transfer to the other ECM. It is also noted that the filter detuning range with the higher SPSC is from −10.4 GHz to −8.2 GHz (the width is ~2 GHz) for η = 0.025, which is evidently wider than that observed in Fig. 13(a). Similarly, the filter detuning range with higher SPSC is from −10.9 GHz to −7.5 GHz (the width is ~3.4 GHz) for η = 0.04, which is significantly narrower than that acquired in Fig. 13(c). In Figs. 13(a) and 13(c), the generated microwave frequencies are around 16.5 GHz and 40GHz respectively. Therefore, large microwave frequencies are more robust to filter detuning.
Figure 15 shows the phase variance as a function of the filter detuning. Same to the linewidth reduction and SPSC observed in Figs. 13(b1) and 13(b4), smaller phase variance is observed in the range from −10.75 GHz to −7.75 GHz (the width is ~3 GHz).
Fig. 15 Phase variance versus the filter detuning. (fi, ξ) = (15 GHz, 0.17), τ = 2.4 ns and η = 0.035.
3.5 Performance of P1 microwave versus the time delay
Figure 16 illustrates the RF spectra of the SL with different feedback delay times in the FOF configuration. Here, (fi, ξ,) = (15 GHz, 0.17), η = 0.025, and (Λ, υ) = (160 MHz, −9 GHz) are used. In Fig. 16(a) with a feedback delay time of τ = 2 ns, the linewidth of the microwave is 37.7 kHz. When the feedback delay time is extended to 4 ns, the linewidth is reduced to 9.5 kHz, as shown in Fig. 16(b). However, the number of SPs in the windows increases and a more evident SP is observed. Further increasing the feedback delay time to 8 ns, the linewidth is reduced further to 5.9 kHz [see Fig. 16(c)]. The number of the SPs in the window is doubled again. In Fig. 16(d) with τ = 16ns, a very narrow linewidth of 4.8 kHz is achieved with more strong SPs. These results clearly demonstrate that a longer feedback delay time leads to a narrower linewidth, which is similar to the results reported in the conventional feedback [34] at the cost of dropping purity of the microwave frequency.
Fig. 16 RF spectra of the SL subject to injection and FOF with a feedback delay time of (a) 2 ns, (b) 4 ns, (c) 8ns and (d) 16 ns.
Figure 17 presents a more detailed investigation of P1 microwave characteristics versus feedback delay time. The feedback strengths are the same as those used in Fig. 13. The top row of Fig. 17 presents the microwave linewidth as a function of feedback delay time. For all three cases, the microwave linewidth reduces with increasing feedback delay time, and the speed of the reduction is slow down gradually and finally the linewidth remains nearly constant at around 1 kHz. It should be noted that the final linewidth could be smaller if the longer time span is adopted. Due to time restraint, the simulation only run over 1 ms. But the trend of linewidth variation as a function of delay time is similar to that with the longer simulation time. The mechanism of the linewidth reduction is that the long delay time leads to dense ECMs, which gives birth to a low phase noise by focusing the energy around the RSCRF. However, the density distribution of ECMs would lead to an increase of the number of ECMs under the profile of the filter passband. Consequently, the ECMs are enhanced and SPs suppression degrades, as observed in Figs. 17(a2)-17(c2). The microwave frequency as a function of the delay time is also investigated in Figs. 17(a3)-17(c3). A little change in the delay time would introduce evident frequency changing. This can be attributed to the fact that the RSCRF opts to shift to its adjacent ECMs [45], and a small change to the delay time would greatly impact the ECMs.
Fig. 17 Linewidth (the first row), SPSC (the second row) and microwave frequency (the third row) of the SL with optical injection and FOF, as a function of the feedback delay time. The parameters (ξ, fi, η) are set as (a) (0.1, 7.5GHz, 0.025), (b) (0.17, 15 GHz, 0.035), (c) (0.365, 31 GHz, 0.04).
To have a better view of the SPs suppression, a map of SPSC as a function of the filter bandwidth and feedback delay time is calculated and presented in Fig. 18. The map shows that the range of feedback delay time for obtaining a SPSC of >20 dB decreases with increasing filter bandwidth. The shorter the feedback delay time, the wider the filter bandwidth that can be used to achieve good SP suppression. Moreover, streaks of SPSC emerging with increasing delay time are also observed. This is because the feedback power exhibits an approximately periodic variation under the filter profile due to the shift of ECMs with respect to the filter.
Fig. 18 Map of SPSC as a function of the filter bandwidth and feedback delay time. (ξ, fi, η) = (0.17, 15 GHz, 0.035).
4. Conclusion
The frequency’s tunability, linewidth and SPs suppression of the P1 microwave generated by a semiconductor laser subject to optical injection and FOF have been systematically investigated. In the FOF configuration, the center frequency of the filter should be set close to the RSCRF. Therefore, the injection parameters and the center detuning should be carefully selected. A broadly tunable narrow linewidth microwave has been demonstrated by adjusting the injection parameters only or by adjusting the injection parameters and filter center frequency. The FOF configuration has a self-adaptive RSCRF that can shift to the center frequency of filter to some range. The range is related to the P1 microwave frequency and the feedback strength and can be extended to several gigahertz. To some extent, filter detuning can also be used to tune the frequency of the P1 microwave. The investigation of the linewidth reduction and phase variance shows that the microwave linewidth and phase variance decrease with increasing feedback strength or feedback delay time within a certain range. Further increasing the feedback strength or feedback delay time has little influence on the linewidth reduction. For very high feedback strength, the SL is also driven out P1 dynamic, linewidth and phase variance increase again with the increase of the feedback strength. Increasing the feedback strength can also benefit the SP suppression but increasing the feedback delay time would degrade SP suppression. Therefore, there is a trade-off between the linewidth reduction and SP suppression with increasing delay time. Moreover, the filter bandwidth also plays a significant role in the SP suppression and the stability of the generated P1 microwave. P1 dynamic can withstand higher feedback strengths in the FOF configuration with a smaller filter bandwidth. Also, better SP suppression can be achieved with a narrower filter bandwidth. The physical mechanism of the SP suppression in the FOF configuration is that the filter can effectively limit the ECMs.
Funding
National Natural Science Foundation of China (61471087, 61671119, 61675154, 61711530652), the China Scholarship Council (201706070007), and the Sêr Cymru National Research Network in Advanced Engineering and Materials.
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