## Abstract

Phase noise of the period-one (P1) nonlinear dynamical oscillation in an optically injected semiconductor laser is numerically investigated. The P1 dynamics causes the laser output intensity to oscillate at a widely tunable frequency for photonic microwave generation, although the intrinsic spontaneous emission in the laser inevitably degrades the microwave signal and manifests as the oscillation phase noise. To characterize the phase noise, the P1 microwave linewidth is first numerically examined through the rate equations with a Langevin term. The P1 microwave linewidth is found to vary with the injection parameters. It is nearly minimized when the microwave power maximizes. Owing to the laser nonlinearities, the P1 microwave linewidth can even be smaller than the free-running optical linewidth. By adding an optical feedback to the laser, the P1 microwave linewidth is found to reduce as the feedback strength and feedback delay increase, in which an inverse-square dependency is followed asymptotically. By modification to a dual-loop feedback, noisy side peaks around the central P1 frequency are effectively suppressed through the Vernier effect. The dual-loop feedback maintains a low phase noise variance over a wide tuning range of the P1 frequency, while allowing long delay times for significant P1 microwave linewidth narrowing.

© 2015 Optical Society of America

## 1. Introduction

Nonlinear dynamical period-one (P1) oscillations in optically injected semiconductor lasers have been actively investigated for photonic microwave generation [1–13]. Photonic microwave generation enables the transmission of microwave signals over optical fibers with no electromagnetic interference and low propagation loss [14–16]. Approaches adopting high-frequency electronic sources for photonic microwave generation include direct modulation on injection-locked lasers [17–19], external modulation using electrooptic materials [20], and side-band locking along with phase-lock loops on modulated light sources [21], which can yield very low phase noise depending on the electronic sources. However, there are also several approaches requiring no high-frequency electronic sources for photonic microwave generation. For example, mode-locked laser diodes (MLLDs) are excellent for monolithically generating photonic microwave tones at the pulse repetition rates. The frequency tunability is limited by the fixed cavity lengths, though some injection and feedback schemes were recently employed to improve the frequency stability and time-bandwidth products in MLLDs [22–26]. Two-section lasers with proper cavity designs had led to pioneering works on self-pulsation phenomena [27–29], which can be observed even without involving any external perturbations. Interesting analysis and modeling of the phase noise associated with the self-pulsations have been reported [30–34]. The optoelectronic oscillators (OEOs) also require no electronic sources to generate microwave signals. Employing long fiber-based feedback loops yield excellent stabilities, though the loops still incorporate high-frequency electronic filters and detectors. Replacing these electronic devices with photonic devices can further improve the tunability and other performances of OEOs [35–39]. Dual-wavelength operation was implemented in fiber lasers to generate microwave frequencies by proper seeding from independent lasers or by different gratings with mechanical or thermal tuning [40, 41]. It was also monolithically implemented in semiconductor lasers with arrayed waveguide gratings to yield discrete tunability [42]. A monolithic approach with three mutually injected lasers was also investigated recently for microwave generation tunable up to 40 GHz, where a clear microwave linewidth reduction to 2.5 MHz was attained by mutual locking through frequency mixing [43].

The nonlinear dynamical P1 oscillation has emerged as an interesting alternative to photonic microwave generation [1, 16]. The P1 oscillation is among a variety of nonlinear dynamics such as period-doubled, quasi-periodic, and chaotic oscillations for lasers under continuous-wave (CW) injections. The P1 oscillation is obtained when a stably locked laser experiences a Hopf bifurcation, which generates a modulation of the laser emission intensity at a microwave frequency [1, 13, 44]. The P1 oscillation has been investigated in conventional single-mode distributed feedback (DFB) lasers [1, 2, 45–47], quantum-dash or quantum-dot lasers [5–7], as well as vertical-cavity surface-emitting lasers (VCSELs) [3, 4, 48, 49]. Dual-wavelength injection and multi-transverse mode injection have also been reported for generating photonic microwave signals [3, 9, 50].

For photonic microwave generation, the nonlinear dynamical P1 oscillation offers the following advantages [16]. The P1 frequency is widely tunable beyond the original relaxation resonance frequency, reaching even the terahertz band [6, 7]. The P1 oscillation generates photonic microwave signals all-optically without using any electronic microwave sources [2, 16]. Single sideband (SSB) optical spectra are attainable by adjusting the injection parameters with the benefit of minimizing the microwave power penalty during transmission through dispersive fibers [10–12, 46]. Intensity modulation depth of nearly 100% can be attained for generating maximal microwave power at a constant optical average power [5, 10]. Data modulation on the microwave signal can be realized by modulating the laser bias current electrically or the injection optically [5, 47]. These advantages were utilized in a number of applications such as radio-over-fiber downlink transmission [5], uplink transmission [46, 47], clock frequency division [51, 52], modulation format conversion [11, 53], photonic microwave amplification [10], and Doppler velocimeter with long detection ranges [54].

In spite of these advantages, the P1 oscillation dynamics inherently contains phase noise due to fluctuations in the lasers. The phase noise causes adverse effects such as reductions of the signal-to-noise ratios in communication links [5], degradations of the noise performance in photonic microwave amplification [10], and limitations of the maximal detection range in Doppler velocimeters [54]. For instance, the velocity resolution degrades within a few meters using a P1 oscillation with a 13.9-MHz microwave linewidth [54]. As a result, some techniques were developed for reducing the phase noise. Experiments have demonstrated double-locking using a microwave source [55, 56], optoelectronic feedback employing an electronic feedback loop [57], and polarization-rotated feedback added with the optoelectronic feedback [2], where a reduced linewidth of about 3 kHz was obtained for P1 oscillation at 6 GHz. To avoid the frequency limitations of high-speed electronics, a dual-loop optical feedback was demonstrated as an all-optical approach with the P1 oscillation frequency increased to over 45 GHz while maintaining a narrow linewidth below 50 kHz [45]. Such a dual-loop feedback on P1 oscillation required no high-speed electronics, demonstrated wide frequency tunability, and significantly reduced both the microwave linewidth as well as the phase variance [45]. However, there were few reports investigating the P1 oscillation noise numerically. Interesting operating points with the P1 oscillation frequency being insensitive to the injection detuning frequency were identified from recent simulations [1]. Data transmission performances were reported by modeling the operational noise of the pump or injection [5]. Nonetheless, the influences of the intrinsic spontaneous emission noise on the P1 oscillation remain to be systematically investigated.

In this paper, the phase noise of the P1 oscillation due to the spontaneous emissions in the injected laser is investigated numerically. As a fundamental noise source intrinsic to the laser, spontaneous emission is modeled by a Langevin fluctuation force in the rate-equation model, causing phase noise in the P1 oscillation, which broadens the P1 microwave linewidth. To begin with, only optical injection is applied to invoke the P1 oscillation. The P1 microwave linewidth is found to vary with the injection parameters. It is minimized when the amplitude of the intensity oscillation maximizes at any given P1 frequency. Interestingly, the P1 microwave linewidth can be even smaller than the free-running optical linewidth, which is impossible without nonlinear interactions in the laser. Moreover, by adding a single-loop optical feedback to the laser, the P1 microwave linewidth is observed to clearly reduce. The linewidth is found as inversely proportional to the square of the feedback strength. It is also found as inversely proportional to the square of the feedback delay time. However, noisy side peaks in the power spectrum are enhanced as the delay lengthens, which prompts the use of a dual-loop optical feedback. The dual-loop optical feedback is found to significantly suppress the side peaks though the Vernier effect when the delay times are incommensurate. So the dual-loop feedback enables microwave linewidth narrowing by adopting long feedback delay times, while keeping a low phase variance regardless of the P1 oscillation frequency. With a free-running optical linewidth of 100 MHz, the laser in P1 oscillation at 24 GHz gives a microwave linewidth of 22 MHz, which is then reduced to 4 kHz and 200 Hz by proper single-loop and dual-loop feedbacks, respectively.

Overall, the numerical results are qualitatively consistent with the previously reported experiments on dual-loop optical feedback [45]. The results also complement the latest report on the laser dynamics with a short single-loop feedback [58], which involves severe mode competition that can degrade the frequency purity of the P1 oscillation. It is also worth mentioning some early investigations of oscillatory nonlinear dynamics with noise modeling in self-pulsating two-section lasers [34]. Following this introduction, the simulation model is presented in Section 2. The P1 oscillation generated by the optical injection without any feedback is investigated in Section 3. The effects of single-loop and dual-loop feedback are studied in Sections 4 and 5, respectively. A discussion in Section 6 is then followed by a conclusion in Section 7.

## 2. Simulation model

Figure 1 shows the schematic for photonic microwave generation using P1 oscillation induced by optical injection with phase noise reduction by optical feedback. The schematic resembles that previously adopted in an experimental work [45]. CW light from a master laser unidirectionally injects into a slave laser through an optical circulator. The slave laser is a single-mode semiconductor laser biased above its threshold. The injection is specified by parameters (*ξ*_{i}, *f*_{i}). The injection strength *ξ*_{i} is a normalized quantity proportional to the injection field, while the injection detuning frequency *f*_{i} is the difference of the injection optical frequency from the free-running optical frequency of the slave laser [13, 59]. The P1 oscillation is invoked in the slave laser when the detuning frequency is above the Hopf bifurcation line in the injection parameter space [60]. The emission of the slave laser carries a modulation at a microwave frequency *f*_{0} due to the P1 oscillation [16]. This generates optical frequency components separated by *f*_{0} from the injected optical frequency in the optical spectrum. Through the circulator, the slave laser emission is converted by a photodetector into a microwave output signal at *f*_{0} [12].

Even with a stable CW injection from the master laser, the slave laser is inherently influenced by its spontaneous emission noise, which gives rise to phase noise of the P1 oscillation. To reduce the phase noise, optical feedback loops are added to the slave laser, as the gray lines in Fig. 1 show. Light from the slave laser passes the circulator into the loops and is then fed back optically to the slave laser. One loop is sufficient to narrow the microwave linewidth of the central peak, though two loops are adopted to suppress the side peaks in the power spectrum [45]. The two loops are specified by the optical feedback parameters (*ξ*_{f1}, *τ*_{1}) and (*ξ*_{f2}, *τ*_{2}), respectively. The feedback strengths *ξ*_{f1} and *ξ*_{f2} are normalized quantities proportional to the feedback fields through the respective loops back into the slave laser, where the total feedback strength *ξ*_{f} = *ξ*_{f1} +*ξ*_{f2} is in practice limited by the optical coupling efficiency from the slave laser into the circulator [44]. The feedback delay times *τ*_{1} and *τ*_{2} are the group delays of the optical signal emitting from and returning to the slave laser through the respective loops [61]. The slave laser, injection, and feedbacks are assumed to have aligned polarizations. The phase differences are ignored as they do not show strong influences on the oscillation noise in the following simulation results.

The dynamics of the slave laser is described by the intracavity optical field amplitude *A*(*t*) and the charge carrier density *N*(*t*). The field amplitude is a complex phasor quantity with respect to the free-running optical frequency of the slave laser. For convenience, the field amplitude is normalized to *a* = *A/|A*_{0} and the charge carrier density is normalized to ñ = (*N* − *N*_{0})*/N*_{0}, where *A*_{0} and *N _{0}* are respectively the free-running values of

*A*and

*N*. The slave laser is modeled by the following rate equations [13, 59]:

*γ*

_{c}is the cavity decay rate,

*γ*

_{s}is the spontaneous carrier relaxation rate,

*γ*

_{n}is the differential carrier relaxation rate,

*γ*

_{p}is the nonlinear carrier relaxation rate,

*b*is the linewidth enhancement factor, and $\tilde{J}$ is the normalized bias current above threshold. The term with

*ξ*

_{i}represents the optical injection [6, 12, 44]. The terms with

*ξ*

_{f1}and

*ξ*

_{f2}represent the optical feedbacks from the two loops [61].

A Langevin fluctuating force *f* is included in Eq. (1) to account for the spontaneous emission noise of the slave laser [59, 62]. Its real and imaginary parts are mutually independent. The Langevin force at different times are also independent, where ergodicity is observed with the following averages:

Here, Δ*ν* is the full width at half-maximum (FWHM) optical linewidth when the slave laser is free-running [63]. Unlike other environmental fluctuations such as temperature and detuning variations, the spontaneous emission noise represented by *f* is intrinsic to the slave laser and cannot be avoided. In the simulations, the FWHM optical linewidth of the free-running slave laser is taken as Δ*ν* = 100 MHz [3]. The parameters from a typical semiconductor laser are adopted [12]: *γ*_{c} = 5.36 *×* 10^{11} s^{−1}, *γ*_{s} = 5.96 *×* 10^{9} s^{−1}, *γ*_{n} = 7.53 × 10^{9} s^{−1}, *γ*_{p} = 1.91 × 10^{10} s^{−1}, *b* = 3.2, and
$\tilde{J}=1.222$. They correspond to a relaxation resonance frequency of *f*_{r} = 10.25 GHz. Second-order Runge-Kutta integration is conducted on Eqs. (1)–(2). The optical spectrum and power spectrum of the slave laser emission are obtained by applying the Fourier transform on *a*(*t*) and *|a*(*t*)|^{2}, respectively. The optical spectrum directly corresponds to the slave laser emission light, whereas the power spectrum corresponds to the microwave output of the photodetector in Fig. 1. The numerical integration has a time step of 0.93 ps. It is carried over a time span of effectively up to 1.05 ms, which is orders of magnitude longer than those reported in most related works [1, 12, 16]. The long time span enables high spectral resolution for examining the narrow microwave linewidths.

## 3. Optical injection-induced P1

The P1 oscillation is invoked in the slave laser by optical injection alone without applying any optical feedback. The optical injection parameters (*ξ*_{i}, *f*_{i}) determine the P1 oscillation frequency *f*_{0}. The optical feedback strengths *ξ*_{f1} and *ξ*_{f2} are set to zero. The variation of the P1 microwave linewidth at different oscillation frequencies is examined in this section. Figures 2(a)–2(d) show the optical spectra and the power spectra of the slave laser emission under P1 oscillations of different frequencies *f*_{0}. The resolution bandwidths of the optical and power spectra are 8 MHz and 0.06 MHz, respectively. The optical spectra are offset to the optical frequency of the free-running slave laser, while the power spectra are centered at the respective P1 oscillation frequencies *f*_{0}. For each value of *f*_{0} in Fig. 2, the injection parameters are chosen to maximize the microwave power, which corresponds to maximizing the intensity modulation depth in the time-domain intensity waveforms. The intensity modulation depth is found to exceed 94% under the injection parameters in Fig. 2. Throughout this work, the microwave power is estimated by integrating the power spectrum over *±*1 GHz around *f*_{0}, where the microwave harmonics at multiples of *f*_{0} are at least 20 dB weaker than the fundamental and are hence ignored.

#### 3.1. Microwave linewidth Δ f_{0} versus frequency f_{0}

In Fig. 2(a) with (*ξ*_{i}, *f*_{i}) = (0.03, 4.0 GHz), the slave laser exhibits P1 oscillation at *f*_{0} = 10.2 GHz. From the optical spectrum in Fig. 2(a-i), the optical injection with detuning frequency *f*_{i} is regenerated in the slave laser as the red arrow indicates. The P1 dynamics gives rise to a number of optical sidebands separated by multiples of the P1 frequency *f*_{0} from the regenerative component. These optical sidebands generate a beat signal at *f*_{0} in the power spectrum, as Fig. 2(a-ii) shows. However, the spontaneous emission noise represented by the Langevin fluctuating force broadens the optical linewidths of the optical sidebands. It also broadens the microwave linewidth at *f*_{0} in the power spectrum, which has a FWHW microwave linewidth of Δ *f*_{0} = 55 MHz based on a Lorentzian fit.

In Fig. 2(b) with (*ξ*_{i}, *f*_{i}) = (0.12, 7.0 GHz), the P1 frequency *f*_{0} increases to 18 GHz. The optical spectrum in Fig. 2(b-i) again contains the regenerative component at *f*_{i} and the optical sidebands separated by multiples of *f*_{0}. The strongest components of the optical spectrum are the regenerative component at *f*_{i} and its first lower sideband at *f*_{i} − *f*_{0}. The two components are of equal amplitude and so attain the maximal intensity modulation depth of 100% [12]. The power spectrum in Fig. 2(b-ii) shows the beat signal at *f*_{0}, which has a narrower microwave linewidth Δ *f*_{0} = 14 MHz. This is actually the narrowest microwave linewidth of P1 oscillation if no feedback is applied. Interestingly, the P1 microwave linewidth Δ *f*_{0} is even narrower than the free-running optical linewidth Δ*ν* of the laser. The narrow linewidth is allowed by the nonlinear frequency mixing among a number of optical frequency components in the P1 optical spectrum of Fig. 2(b-i). The P1 oscillation is not exactly the same as heterodyning the injection light with a free-running slave laser [12,13]. Similar narrowing of microwave linewidth through nonlinear frequency mixing was also reported in mutually coupled three-laser systems with DFB lasers as well as semiconductor ring lasers (SRLs) [43, 64].

In Fig. 2(c) with (*ξ*_{i}, *f*_{i}) = (0.17, 14.2 GHz), the P1 frequency *f*_{0} increases to 24 GHz. The associated microwave linewidth slightly degrades to Δ *f*_{0} = 22 MHz. In Fig. 2(d) with (*ξ*_{i}, *f*_{i}) = (0.48, 55.5 GHz), the P1 frequency *f*_{0} continually increases to 60 GHz, which is much greater than the relaxation resonance frequency *f*_{r} of the laser. From the optical spectrum in Fig. 2(d-i), the components at *f*_{i} and *f*_{i} *− f*_{0} are now much more dominating, in which they are at least 30 dB stronger than the rest of the sidebands. According to a two-wavelength approximation, the component at *f*_{i} is simply the regeneration of the emission, while the component at *f*_{i} − *f*_{0} is the emission of the slave laser near its cavity resonance [13]. The slave laser is different from being free-running because its cavity resonance is red-shifted by the injection. Strong injection can suppress the emission power at *f*_{i} *− f*_{0} and thereby can increase the optical linewidth at *f*_{i} *− f*_{0} beyond Δ*ν* [63]. Through beating with the regenerative component, the component at *f*_{i} *− f*_{0} produces the power spectrum in Fig. 2(d-ii). The microwave linewidth is increased to Δ *f*_{0} = 105 MHz, which is now larger than Δ*ν*. For maximal microwave power, further increasing *f*_{0} requires increasing *f*_{i}. It is associated with reducing the emission power at *f*_{i} − *f*_{0} with further degradation of Δ *f*_{0}. In short, Fig. 2 illustrates the wide tunability of the P1 frequency *f*_{0} beyond the relaxation resonance frequency *f*_{r}, though the associated microwave linewidth Δ *f*_{0} varies with the injection parameters.

#### 3.2. Microwave linewidth Δ f_{0} versus injection parameters (ξ_{i}, f_{i})

To systematically investigate the behavior of the P1 microwave linewidth Δ *f*_{0}, Fig. 3 shows its contour map in the injection parameter space (*ξ*_{i}, *f*_{i}). The large blue region above the Hopf bifurcation line is a region of P1 oscillation. The blue contour map indicates the microwave linewidth Δ *f*_{0} of the P1 oscillation. At (*ξ*_{i}, *f*_{i}) = (0.12, 7.0 GHz), the yellow point in Fig. 3 gives the minimal linewidth of Δ *f*_{0} = 14 MHz. Such a minimal-linewidth point is attained at *f*_{0} = 18 GHz, as Fig. 2(b) shows. Then, when *ξ*_{i} goes to zero or when *f*_{i} increases much beyond *f*_{r}, Δ *f*_{0} increases up to around Δ*ν* in Fig. 3. This is because the low-strength or largely detuned injection does not cause much nonlinear frequency mixing in the slave laser, so that the P1 oscillation reduces to a simple heterodyne process between the regenerated injection and the nearly free-running slave laser. However, when the Hopf bifurcation line in Fig. 3 is crossed, the slave laser becomes stably locked by the injection. The P1 oscillation becomes damped such that its linewidth quickly increases near the Hopf bifurcation line. Moreover, the slave laser also experiences period-doubling into other oscillatory states in the small white regions with (*ξ*_{i}, *f*_{i}) at around (0.08, 15 GHz) and (0.04, 8 GHz). These regions do not give P1 oscillation and need to be avoided in photonic microwave generation [12], where linewidth broadening at period-doubling bifurcations has been reported [65].

Within the region of P1 oscillation in blue, Fig. 3 also shows the gray contour lines of the P1 oscillation frequency *f*_{0}, which is widely and continuously tunable beyond the relaxation resonance frequency *f*_{r} as the injection parameters vary. Adjusting (*ξ*_{i}, *f*_{i}) along any of these contour lines keeps *f*_{0} constant, while the associated microwave power varies. The dashed line in Fig. 3 indicates the positions of maximal microwave power at any given *f*_{0}. Comparing the dashed line with the map of Δ *f*_{0} reveals that, for any fixed *f*_{0}, the microwave linewidth is nearly minimized as the microwave power maximizes. For example, the minimal-linewidth point with Δ *f*_{0} = 14 MHz lies directly on the dashed line in Fig. 3 for *f*_{0} = 18 GHz. Further increasing *f*_{0} along the dashed line is associated with increasing Δ *f*_{0}.

Interestingly, there are recent works focusing on points with the contour lines of *f*_{0} being tangentially vertical, which ensure insensitivity of *f*_{0} to variations of *f*_{i} [1,2]. Such a detuning-insensitive point for *f*_{0} = 18 GHz is at (*ξ*_{i}, *f*_{i}) = (0.125, 3.5 GHz) according to Fig. 3. However, the detuning-insensitive point is not the same as the minimal-linewidth point. This causes no contradiction because the intrinsic spontaneous emission noise, rather than the detuning variation, is considered as the fundamental source of fluctuation here in Fig. 3. As for experimental investigations, the temperature fluctuations have not been totally eliminated thus far, so both detuning variations and spontaneous emissions contribute to the linewidth. The linewidth, nonetheless, was experimentally confirmed to vary with the injection parameters [2, 45].

Summarizing this section on optical injection, the P1 oscillation can have a microwave linewidth Δ *f*_{0} being smaller than the free-running optical linewidth Δ*ν*. Given a P1 microwave frequency *f*_{0}, its linewidth Δ *f*_{0} is nearly minimized when the microwave power maximizes. The P1 frequency *f*_{0} can also be tuned much beyond *f*_{r}, but it is associated with increasing Δ *f*_{0}. In order to reduce the microwave linewidth and phase noise, optical feedbacks on the P1 oscillation are considered in the following sections.

## 4. P1 with single-loop feedback

Linewidth reduction through single-loop optical feedback is considered in this section on the optically injected laser in P1 oscillation. Similar to a solitary laser without injection [66,67], the feedback loop stores the phase information, sets up external cavity modes, and causes reduction of the linewidth for the central peak along with residual side peaks separated by the reciprocal of the feedback delay time. In this section, the optical injection parameters are fixed at (*ξ*_{i}, *f*_{i}) = (0.17, 14.2 GHz). The laser enters P1 oscillation at *f*_{0} = 24 GHz as in Fig. 2(c) prior to applying any feedback. To avoid losing generality, the injection parameters are chosen not to correspond to the minimum-linewidth point in Fig. 3 as the feedback is found to reduce the microwave linewidth regardless of the P1 frequency. Only one loop in Fig. 1 is turned on with feedback parameters (*ξ*_{f1}, *τ*_{1}) denoted simply as (*ξ*_{f}, *τ*), whilst the other loop remains off by keeping *ξ*_{f2} at zero. Figures 4(a)–4(d) show the optical and power spectra of the laser emission at different feedback strengths. The feedback delay time is fixed at *τ* = 2.4 ns. The optical spectra are again offset to the free-running frequency of the laser. The power spectra are centered at 24 GHz. The resolution bandwidth of the optical spectra remains at 8 MHz, but that of the power spectra is decreased to 1 kHz for examining the narrow central peaks. The high-resolution spectra are made possible by the significantly increased effective simulation time-span compared to most earlier works [1,12,16].

In Fig. 4(a) with *ξ*_{f} = 0.010, the slave laser continues exhibiting P1 oscillation at *f*_{0} = 24 GHz. The optical spectrum in Fig. 4(a-i) with feedback is essentially the same as that in Fig. 2(c-i) without feedback. The regenerative component indicated by the red arrow and the optical sidebands separated by multiples of *f*_{0} are again identified. However, comparing with Fig. 2(c-ii), the power spectrum in Fig. 4(a-ii) shows a significant narrowing of the central peak at *f*_{0}. From the original Δ*f*_{0} = 22 MHz, the feedback significantly reduces the FWHM microwave linewidth down to
$\mathrm{\Delta}{{f}^{\prime}}_{0}=93\phantom{\rule{0.2em}{0ex}}\text{kHz}$. Although the feedback also invokes some side peaks equally separated from the central peak by multiples of 1*/τ*, these side peaks are much weaker than the central peak and so do not contribute to the FWHM linewidth. In Fig. 4(b), by increasing the feedback strength to *ξ*_{f} = 0.026, the microwave linewidth is further reduced. The power spectrum of Fig. 4(b-ii) shows the microwave linewidth narrowing to
$\mathrm{\Delta}{{f}^{\prime}}_{0}=9\phantom{\rule{0.2em}{0ex}}\text{kHz}$ for the central peak, where the magnitudes of the side peaks are suppressed as well. The optical spectrum in Fig. 4(b-i) contains the same frequency components as before. Though the optical background noise is slightly enhanced in between the frequency components, it remains much weaker than the two strongest frequency components at *f*_{i} and *f*_{i} − *f*_{0} by over 30 dB.

In Fig. 4(c), the feedback strength increases to *ξ*_{f} = 0.034. The power spectrum of Fig. 4(c-ii) shows that the microwave linewidth of the central peak further reduces to
$\mathrm{\Delta}{{f}^{\prime}}_{0}=4\phantom{\rule{0.2em}{0ex}}\text{kHz}$, which is over three orders of magnitude below Δ*f*_{0} prior to applying any feedback. From the optical spectrum in Fig. 4(c-i) the regenerative component at *f*_{i} and the sideband at *f*_{i} − *f*_{0} are still the two strongest components, yet subharmonic components emerge in between the original components because of proximity to a period-doubling bifurcation [52]. A subharmonic signal at *f*_{0}/2 emerges accordingly in the power spectrum, but it is over 9 dB weaker than the fundamental at *f*_{0} and can be easily filtered out by microwave filters in practical systems.

Obviously, the feedback strength cannot be increased indefinitely. For instance, in Fig. 4(d), the slave laser enters into chaotic oscillation when the feedback strength increases to 0.060. The optical spectrum in Fig. 4(d-i) still contains the regenerative component at *f*_{i}, but the spectrum is much broadended into a continuum of chaos. The power spectrum is broadened significantly with only a small portion shown in Fig. 4(d-ii), which contains a repeating pattern in every 1/*τ* as a characteristic of feedback-induced chaos [61]. It is worth mentioning that the different nonlinear dynamical regimes of the laser under both optical injection and feedback were recently studied in detail [8, 58], where chaotic oscillations must be avoided for microwave generation by limiting the maximal feedback strength.

According to the numerical results in Fig. 4, the P1 microwave linewidth reduces as the feedback strength increases as long as the laser remains in periodic oscillation. Experimentally, the feedback-induced narrowing of the P1 microwave linewidth was confirmed [45]. Analytically, the feedback-induced microwave linewidth reduction has not been thoroughly understood. In order to estimate the reduced microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$, it is first observed from Figs. 4(a-i)–4(c-i) that the two components at *f*_{i} and *f*_{i} − *f*_{0} dominate the P1 optical spectra. The regenerative component at *f*_{i} comes from a CW injection and carries nearly no linewidth. So the P1 microwave linewidth is directly translated from the optical linewidth at *f*_{i} − *f*_{0}. Then, by borrowing the results on feedback-induced optical linewidth narrowing in a solitary laser without injection [44,66,67], the feedback-induced optical linewidth narrowing at *f*_{i} − *f*_{0} for the injected laser is estimated. As a result, the reduced microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ for the P1 oscillation with feedback (*ξ*_{f}, *τ*) is approximated as

*f*

_{0}is the microwave linewidth obtained under optical injection alone prior to applying any feedback and the feedback is assumed not to affect the optical power at

*f*

_{i}−

*f*

_{0}as Figs. 4(a-i)–4(c-i) suggest. The factor

*β*depends on the excited external cavity mode [44]. It is equal to unity for the minimum threshold mode and equal to $\sqrt{1+{b}^{2}}$ for the minimum linewidth mode. The latter case sets the lower-bound of $\mathrm{\Delta}{{f}^{\prime}}_{0}$. The dependencies of $\mathrm{\Delta}{{f}^{\prime}}_{0}$ on the feedback parameters are simulated next for verifying Eq. (6).

#### 4.1. Reduced microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ versus feedback strength ξ_{f}

The closed symbols in Fig. 5 show the reduced microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ as the feedback strength *ξ*_{f} varies. The feedback delay time is fixed at *τ* = 2.4 ns. The linewidth is obtained at the P1 frequency of *f*_{0} = 24 GHz because the injection is fixed at (*ξ*_{i}, *f*_{i}) = (0.17, 14.2 GHz). The P1 oscillation has the original microwave linewidth of Δ*f*_{0} = 22 MHz prior to application of the feedback as shown in Fig. 2(c-ii). As the feedback strength *ξ*_{f} increases, the microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ decreases continuously. It reaches a minimal value of 4 kHz at *ξ*_{f} = 0.034 in Fig. 5. Further increasing the feedback strength eventually drives the laser into chaos and so broadens the linewidth.

Using Eq. (6),
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ is estimated as the blue and red curves for *β* = 1 and
$\sqrt{1+{b}^{2}}$, respectively. Provided that the laser remains in periodic oscillation, the simulation results are above the analytical lower-bound as indicated by the red curve. The simulation results are also in close agreement with the analytical results in blue. According to Eq. (6), when the feedback is sufficiently strong with
${\xi}_{\mathrm{f}}\tau \gg {\gamma}_{\mathrm{c}}^{-1}$, the reduced microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ is proportional to
${\xi}_{\mathrm{f}}^{-2}$.

The microwave power of the signal at *f*_{0} is plotted as open symbols in Fig. 5. The power varies by less than 3 dB when *ξ*_{f} < 0.04 as long as the laser remains in periodic oscillation. Therefore, unlike conventional oscillators such as the OEOs, the feedback in Fig. 1 is not responsible for generating the P1 oscillation. The feedback merely stabilizes the fluctuations of the P1 oscillation [57].

#### 4.2. Reduced microwave linewidth $\mathrm{\Delta}{{f}^{\prime}}_{0}$ versus feedback delay time τ

Figure 6 shows a continual reduction of
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ as the feedback delay time *τ* increases. The feedback strength is kept constant at *ξ*_{f} = 0.010. The simulation results are in closed symbols. The analytical estimations of Eq. (6) are again plotted in blue and red with *β* = 1 and
$\sqrt{1+{b}^{2}}$, respectively. The simulated results are clearly bounded from below by the red curve. They are also in good agreement with the analytical results in blue. When the delay is sufficiently long with
${\xi}_{\mathrm{f}}\tau \gg {\gamma}_{\mathrm{c}}^{-1}$, the reduced linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ is proportional to *τ*^{−2}. The inverse-squared dependence of the linewidth on the feedback time is commonly observed, for example, in OEOs as well as in lasers at the Schawlow-Townes limit [39]. Experiments on frequency stabilization of the P1 oscillation were demonstrated with fiber loops [45], which can conveniently realize long delays with
${\xi}_{\mathrm{f}}\tau \gg {\gamma}_{\mathrm{c}}^{-1}$.

For completeness, it is worth mentioning that a short delay with
${\xi}_{\mathrm{f}}\tau ~{\gamma}_{\mathrm{c}}^{-1}$ gives only a few external cavity modes [44]. The modes are far-separated by 1/*τ* in causing severe frequency-shifts of the central microwave peak, which lead to drastic fluctuations of the microwave linewidth exceeding its original value without feedback [58]. Thus, for yielding phase noise reduction, short delays should be avoided. A long delay should instead be adopted according to Fig. 6.

#### 4.3. Phase variance versus feedback parameters (ξ_{f}, τ)

The P1 oscillation phase noise improvement is illustrated through measuring the reduction of the microwave linewidth in Figs. 5 and 6. However, according to Figs. 4(a-ii)–4(c-ii), the central peak in the power spectrum is accompanied by some noisy side peaks separated by 1/*τ*. Both the non-zero linewidth and the side peaks contribute to the P1 oscillation phase noise, which can be quantified by its variance in Fig. 7. The phase noise variance is estimated by integrating the averaged single sideband of the power spectrum centered at *f*_{0}, with normalization to the microwave power, over an offset from *f*_{L} = 3 MHz to *f*_{H} = 1 GHz [22,46,52,56,58]. The phase variance is proportional to the integrated mean square timing jitter of the P1 oscillation [68]. Figure 7 is a contour map of the phase variance in the feedback parameter space (*ξ*_{f}, *τ*). The minimum phase variance of 0.001 rad^{2} is attained at an optimal point (*ξ*_{f}, *τ*) = (0.034, 2.4 ns), which corresponds to Fig. 4(c-ii) with a linewidth of
$\mathrm{\Delta}{{f}^{\prime}}_{0}=4\phantom{\rule{0.2em}{0ex}}\text{kHz}$. On one hand, for a fixed *τ*, the phase variance and the linewidth have similar dependencies on *ξ*_{f}. When *ξ*_{f} increases from zero in Fig. 7, the phase variance first experiences a progressive reduction due to central linewidth reduction and side peak suppression, but then it quickly rises up as the feedback becomes sufficiently strong to drive the laser into chaos. For example, at *τ* = 2.4 ns, the phase variance reduces progressively as *ξ*_{f} increases up to 0.034 and then raises abruptly as the laser becomes chaotic. The linewidth in Fig. 5 follows a similar dependency on *ξ*_{f}. On the other hand, for a fixed *ξ*_{f}, the phase variance and the linewidth do not have the same dependency on *τ*. As *τ* increases, the linewidth decreases monotonically in Fig. 6, while the phase variance does not. For example, at *ξ*_{f} = 0.034, the contour map in Fig. 7 first shows a reduction of the phase variance until *τ* = 2.4 ns, corresponding to the spectrum in Fig. 4(c-ii). However, as *τ* further increases, the side peaks continue to move towards the central peak and increase in magnitude. This causes substantial degradation of the phase variance for large *τ* in Fig. 7. Comparing Figs. 6 and 7, it is clear that a continual increase of *τ* progressively reduces the linewidth, but this eventually causes increment of the phase variance. The side peaks need to be suppressed by modifying the feedback in the next section.

To summarize, this section illustrates the phase noise improvement of the P1 oscillation through adopting the single-loop optical feedback. The reduced microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ is observed to roughly follow Eq. (6). The linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ is asymptotically proportional to
${\xi}_{\mathrm{f}}^{-2}$ when the feedback strengthens until the laser becomes chaotic. The linewidth is also asymptotically proportional to *τ*^{−2} when the feedback delay lengthens, although the side peaks grow in degrading the phase noise variance at long delays.

## 5. P1 with dual-loop feedback

The previous section illustrates the inverse-square dependence of the P1 linewidth on the feedback time. A long delay time is preferred for linewidth narrowing, but the noisy side peaks grow in magnitude as the delay time increases. For the single-loop feedback, the minimum phase variance of 0.001 rad^{2} is obtained at (*ξ _{f}*,

*τ*) = (0.034, 2.4 ns) in Fig. 7, corresponding to the spectrum in Fig. 4(c-ii) with a P1 microwave linewidth of $\mathrm{\Delta}{{f}^{\prime}}_{0}=4\phantom{\rule{0.2em}{0ex}}\text{kHz}$. To further reduce the linewidth while keeping the small phase variance, the dual-loop feedback is considered in this section. The two loops have slightly different delay times so that their side peaks are misaligned in enabling the Vernier effect. The feedback parameters (

*ξ*

_{f1},

*τ*

_{1}) and (

*ξ*

_{f2},

*τ*

_{2}) specify the dual-loop feedback in Fig. 1. The total feedback strength is

*ξ*

_{f}=

*ξ*

_{f1}+

*ξ*

_{f2}. The average delay time is denoted as

*τ*= (

*τ*

_{1}+

*τ*

_{2})/2. The feedback delay time difference is notated

*τ*

_{21}=

*τ*

_{2}−

*τ*

_{1}. Figure 8 shows the results when the feedback delay times are increased to

*τ*

_{1}= 17 ns and

*τ*

_{2}= 20 ns, which are much longer than the original short delay of 2.4 ns in Fig. 4. Beginning with Fig. 8(a), only the loop of

*τ*

_{1}is switched on with

*ξ*

_{f1}= 0.010 and

*ξ*

_{f2}= 0. The feedback with the long delay leads to the reduced central linewidth of $\mathrm{\Delta}{{f}^{\prime}}_{0}=2.7\phantom{\rule{0.2em}{0ex}}\text{kHz}$. Similarly, in Fig. 8(b), only the loop of

*τ*

_{2}is switched on with

*ξ*

_{f2}= 0.010 and

*ξ*

_{f1}= 0. The long feedback delay time again gives a small $\mathrm{\Delta}{{f}^{\prime}}_{0}$ of 2.6 kHz. The long delay times in Figs. 8(a)–8(b) yield the linewidths being much smaller than the linewidth of 93 kHz in Fig. 4(a-ii) for the original short delay time. However, the long delay times in Figs. 8(a)–8(b) also lead to closely packed side peaks separated by 1/

*τ*

_{1}and 1/

*τ*

_{2}, respectively. The side peaks are close to the central peak and so are much stronger than those in Fig. 4(a-ii) for the original delay. In other words, for feedbacks with only one loop, a long delay time is preferred for central linewidth narrowing, but it is associated with strong noisy side peaks that degrade the phase variance.

Because (*τ*_{1}, *τ*_{2}) are basically incommensurate, the side peaks in Figs. 8(a)–8(b) are not perfectly aligned. When both loops are switched on in Fig. 8(c), the misalignment results in a clear suppression of the side peaks as a Vernier effect [24, 69], which brings down the phase variance significantly. The dual-loop feedback is simulated with two equally strong feedback strengths of *ξ*_{f1} = *ξ*_{f2} = 0.005 such that the total feedback strength *ξ*_{f} is still 0.010. As a result, the central linewidth is maintained at
$\mathrm{\Delta}{{f}^{\prime}}_{0}=2.7\phantom{\rule{0.2em}{0ex}}\text{kHz}$, but the phase variance is brought back down to the order of 0.001 rad^{2}. By increasing both feedback strengths to *ξ*_{f1} = *ξ*_{f2} = 0.017, Fig. 8(d) shows a further reduction of the central peak linewidth down to just
$\mathrm{\Delta}{{f}^{\prime}}_{0}=200\phantom{\rule{0.2em}{0ex}}\text{Hz}$ according to a Lorentzian fit, where the phase noise variance is less than 0.001 rad^{2}. Hence, the dual-loop feedback is preferred over single-loop feedback because it maintains a low phase variance while the central linewidth narrows.

The spectra with dual-loop feedback in Fig. 8(c)–8(d) contains residual side peaks separated at multiples of the reciprocal of the average delay *τ*. The magnitudes of the side peaks also follow an undulating envelope at the dual-loop beat frequency 1/*τ*_{21} = 0.33 GHz. Nonetheless, these side peaks are much weaker than the central peak because 1/*τ*_{21} is much greater than the original P1 microwave linewidth of Δ*f*_{0} = 22 MHz without feedback, as in Fig. 2(c-ii). The linewidth reduction and side peaks suppression in Fig. 8 are qualitatively consistent with the reported experimental results [45].

#### 5.1. Phase variance versus total feedback strength ξ_{f}

To systematically quantify the suppression of side peaks, Fig. 9 shows the phase noise variance as the total feedback strength *ξ*_{f} varies. The feedback delay times are kept constant at *τ*_{1} = 17 ns and *τ*_{2} = 20 ns. Generally, the phase variance comprises of the contribution from the Lorentzian tail of the central peak and the contribution from the side peaks. When the side peaks are completely eliminated, the ideal phase variance can be calculated by integrating the Lorentzian tail of the central peak with linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$. Such an ideal phase variance is given by
$\mathrm{\Delta}{{f}^{\prime}}_{0}({f}_{\mathrm{L}}^{-1}-{f}_{\mathrm{H}}^{-1})/2\pi $ when
$\mathrm{\Delta}{{f}^{\prime}}_{0}\ll {f}_{\mathrm{L}}$ _{[22,68]}. Here,
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ is estimated by substituting the total feedback strength *ξ*_{f} and the averaged delay time *τ* into Eq. (6) with *β* = 1. The resultant estimation of the ideal phase variance is shown in Fig. 9 as the blue curve, which is always below the actual phase variance due to incomplete elimination of side peaks.

For single-loop feedback, only one loop is switched on at a time. The up-triangles are from switching on only the loop of *τ*_{1} at strength *ξ*_{f1} = *ξ*_{f} so that *ξ*_{f2} is zero. Likewise, the down-triangles are from switching on only the loop of *τ*_{2} at strength *ξ*_{f2} = *ξ*_{f} so that *ξ*_{f1} is zero. Regardless of which loop is switched on, the phase variance is always much greater than its ideal value in Fig. 9, which is due to the strong side peaks obtained at the long feedback delay times. The triangles also show the gradual reduction of phase variances as *ξ*_{f} increases, although the reduction slope is less than that of the ideal phase variance. The phase variance eventually rises at *ξ*_{f} = 0.024 or 0.014 for the two independent loops as the laser exhibits complex dynamics.

By contrast, the dual-loop feedback yields the phase variance as the circles indicate in Fig. 9. Both loops are switched on with equal strengths such that *ξ*_{f1} = *ξ*_{f2} = *ξ*_{f}/2. The phase variance from dual-loop feedback is always much smaller than that from single-loop feedback because of the suppression of the side peaks. It also reduces more quickly as the feedback strength *ξ*_{f} increases. The laser remains in P1 oscillation over a larger range of *ξ*_{f} for the dual-loop feedback as compared with the single-loop feedbacks. Until the complex dynamics emerges at *ξ*_{f} of over 0.04, the dual-loop feedback is useful for reducing the phase variance down to less than 0.001 rad^{2}. Therefore, although it does not completely eliminate all side peaks, the dual-loop feedback always gives a lower phase variance compared to that from the single-loop feedbacks and widens the range of the total feedback strength for P1 oscillation.

#### 5.2. Phase variance versus feedback strengths (ξ_{f1}, ξ_{f2})

The effect of varying the power distribution between the two loops is considered in Fig. 10, where the phase variance is plotted against the ratio of the feedback strengths *ξ*_{f1} and *ξ*_{f2}. Regardless of the distribution of optical power feeding back through the two loops in Fig. 1, the maximal total feedback strength *ξ*_{f} is limited due to the finite optical coupling efficiency from the slave laser. So Fig. 10 keeps the total feedback strength *ξ*_{f} = *ξ*_{f1} +*ξ*_{f2} constant at 0.010, 0.020, and 0.035 for the triangles, squares, and circles, respectively. The feedback delay times of the two loops remain fixed at *τ*_{1} = 17 ns and *τ*_{2} = 20 ns. For any *ξ*_{f}, Fig. 10 clearly shows minimization of the phase variance when the ratio *ξ*_{f1}:*ξ*_{f2} is optimized to unity. Such a feedback with equal strengths from the two loops corresponds to a filter of maximal extinction ratio, which results in effective suppression of the side peaks. In addition, as the ratio *ξ*_{f1}:*ξ*_{f2} deviates from unity at a fixed *ξ*_{f}, the dual-loop feedback becomes progressively similar to a single-loop feedback such that the side peaks become less suppressed, where the laser eventually enters other dynamical states. For instance, at *ξ*_{f} = 0.035, the laser remains in P1 oscillation only when *ξ*_{f1}:*ξ*_{f2} is between 0.1 and 10. Therefore, it is optimal to distribute the feedback power equally among the two loops for P1 oscillation phase variance minimization.

#### 5.3. Phase variance versus feedback delay times (τ_{1}, τ_{2})

The effect of varying the feedback delay times of the two loops is considered in Fig. 11. The phase variance is shown as a map of (*τ*_{1}, *τ*_{2}). The two loops are set at equal feedback strengths of *ξ*_{f1} = *ξ*_{f2} = 0.010. The region with *τ*_{1} > *τ*_{2} is not presented since the roles of the two delays are interchangeable. Over most part of the map with delays reaching 40 ns in Fig. 11, the dual-loop feedback successfully suppresses the phase variance to the order of 0.001 rad^{2}, which is the minimum value attainable by single-loop feedback at a much shorter delay of 2.4 ns in Fig. 7. There are also many white regions scattered in Fig. 11, which represent phase variances of even less than 0.001 rad^{2}. However, straight lines of relatively large phase variances emerge when *τ*_{1} and *τ*_{2} become commensurate. Lines with *τ*_{1}:*τ*_{2} = 1:1, 1:2, 2:3, and 3:1 are clearly identified in Fig. 11. These rational values of *τ*_{1}:*τ*_{2} result in strong noisy side peaks at the common multiples of the reciprocals of the delays. Moreover, at any fixed value of *τ*_{1}:*τ*_{2}, the phase variance generally increases as the delay times increase. This is due to the enhancement of the residual side peaks as their separation 1/*τ* reduces. As the dual-loop is reduced to a single-loop when *τ*_{1}:*τ*_{2} = 1:1, a comparison of the line along *τ*_{1}:*τ*_{2} = 1:1 with the rest of the map confirms the superiority of dual-loop feedback over the single-loop feedback as long as commensurate (*τ*_{1}, *τ*_{2}) are avoided.

Summarizing this section, the dual-loop feedback clearly suppresses the side peaks through the Vernier effect. It allows using long delay times to reduce the P1 linewidth while maintaining a low phase variance. The two loops are preferably of equal strengths with incommensurate delays.

## 6. Discussion

Up to this point, the results focus on the linewidth narrowing for the P1 oscillation at *f*_{0} = 24 GHz by optical injection of (*ξ*_{i}, *f*_{i}) = (0.17, 14.2 GHz). From a free-running optical linewidth of Δ*ν* = 100 MHz, the P1 oscillation of microwave linewidth Δ*f*_{0} = 22 MHz is shown in Fig. 2(c). The single-loop feedback then reduces the microwave linewidth to
$\mathrm{\Delta}{{f}^{\prime}}_{0}=4\phantom{\rule{0.2em}{0ex}}\text{kHz}$ at (*ξ*_{f}, *τ*) =(0.034, 2.4 ns) in Fig. 4(c-ii). The dual-loop feedback further reduces the linewidth to merely
$\mathrm{\Delta}{{f}^{\prime}}_{0}=200\phantom{\rule{0.2em}{0ex}}\text{Hz}$ at (*τ*_{1}, *τ*_{2}) = (17 ns, 20 ns) in Fig. 8(d), while the phase variance is maintained below 0.001 rad^{2}.

For completeness, the effects of the single- and dual-loop feedbacks are also investigated as the P1 oscillation frequency *f*_{0} and the free-running laser linewidth Δ*ν* are varied in Fig. 12. The phase variance is obtained without feedback, with single-loop feedback, and with dual-loop feedback for the triangles, squares, and circles, respectively. The total feedback strength is kept constant at *ξ*_{f} = 0.010. The single-loop delay time is *τ* = 17 ns. The dual-loop delay times are (*τ*_{1}, *τ*_{2}) = (17 ns, 20 ns) with equally distributed feedback strengths.

In Fig. 12(a), the phase variance is shown as the P1 oscillation frequency *f*_{0} varies, where the injection parameters are chosen along the dashed line of maximal microwave power in Fig. 3. A wide tuning range of *f*_{0} from around the relaxation resonance frequency of *f*_{r} = 10.25 GHz to over 60 GHz is shown. Increasing *f*_{0} is associated with a slight reduction of the microwave power within about 4 dB [12,46]. The phase variance is found to improve by one to two orders of magnitude using the single-loop feedback. Another order of magnitude of improvement is then obtained by the suppression of side peaks through the dual-loop feedback. Although the use of the sideband power ratio for the phase variance estimation becomes less accurate for the large linewidths without feedback, phase noise reduction by single- and dual-loop feedback independent of the P1 frequency is qualitatively consistent with recent experiments [45]. Also, regardless of whether any feedback is applied, Fig. 12(a) shows a sharp increase of the phase variance as *f*_{0} reduces in approaching *f*_{r}. This is due to the proximity to the emergence of chaotic or other nonlinear dynamics. In Fig. 12(b), the impact of the free-running linewidth Δ*ν* of the slave laser is illustrated. Without any feedback, the P1 phase variance is found as directly proportional to Δ*ν*, where the phase variance reduction by the single- and dual-loop feedbacks are independent of Δ*ν*. Hence, the phase noise improvement by optical feedback is generally applicable over wide ranges of P1 frequency as well as laser linewidth.

## 7. Conclusion

In conclusion, the nonlinear dynamical P1 oscillation of a semiconductor laser subject to both optical injection and feedback is systematically investigated by simulation with the Langevin noise. With optical injection alone, the P1 oscillation frequency *f*_{0} is widely tunable beyond the relaxation resonance frequency *f*_{r}, where the microwave linewidth Δ*f*_{0} varies with the injection parameters. By the P1 nonlinear dynamics, the microwave linewidth Δ*f*_{0} can even be smaller than the free-running optical linewidth Δ*ν*. By adding a single-loop optical feedback, the reduced microwave linewidth
$\mathrm{\Delta}{{f}^{\prime}}_{0}$ is found as asymptotically proportional to
${\xi}_{\mathrm{f}}^{-2}{\tau}^{-2}$. By modifying to a dual-loop feedback, the Vernier effect of incommensurate delay times effectively suppresses the side peaks in the power spectrum, which enables P1 linewidth narrowing at long delay times while maintaining a low phase noise variance. The results support analytical and experimental investigations on the all-optical narrow-linewidth tunable photonic microwave generation by the laser nonlinear dynamics.

## Acknowledgments

The authors thank Dr. Marc Sorel for valuable discussions. This work was supported in part by a grant from the Research Grant Council of Hong Kong under Project CityU 110712, a grant from the City University of Hong Kong under Project 7002726, and a grant from the Royal Society U.K. under Project IE120157.

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