## Abstract

This study numerically investigates the enhancement of photonic microwave generation using an optically injected semiconductor laser operating at period-one (P1) nonlinear dynamics through ultrashort optical feedback. For the purpose of practical applications where system miniaturization is generally preferred, a feedback delay time that is one to two orders of magnitude shorter than the relaxation resonance period of a typical laser is emphasized. Various dynamical states that are more complicated than the P1 dynamics can be excited under a number of ultrashort optical feedback conditions. Within the range of the P1 dynamics, on one hand, the frequency of the P1 microwave oscillation can be greatly enhanced by up to more than three folds. Generally speaking, the microwave frequency enhances with the optical feedback power and phase, while it varies saw-wise with the optical feedback delay time. On the other hand, the purity of the P1 microwave oscillation can be highly improved by up to more than three orders of magnitude. In general, the microwave purity improves with the optical feedback power and delay time, while it only varies within an order of magnitude with the optical feedback phase. These results suggest that the ultrashort optical feedback provides the optically injected laser system with an extra degree of freedom to manipulate/improve the characteristics of the P1 microwave oscillation without changing the optical injection condition.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Microwaves have found their importance in numerous application areas, such as wireless access networks, radars and sensors, satellite communication, and warfare systems, and have therefore continuously attracted much research interest in their generation, processing, and distribution. For microwave generation, photonic approaches provide various promising advantages over their electronic counterparts, such as capability of microwave generation over the millimeter-wave band, broad and continuous microwave frequency tunability, and long-distance microwave distribution through optical fibers [1, 2]. These photonic approaches take advantage of different generation mechanisms, which include optical heterodyne between two independent lasers [3, 4], mode-locked semiconductor lasers [5, 6], optoelectronic oscillators [7, 8], direct modulation of injection-locked lasers [9, 10], and cascade of external modulation [11, 12].

Lately, a photonic microwave generation approach based on optically injected semiconductor lasers operating at period-one (P1) nonlinear dynamics has received considerable attention [13–27]. The P1 dynamics can be excited by undamping the relaxation resonance of a semiconductor laser through continuous-wave optical injection beyond a Hopf bifurcation [28, 29]. From the spectral perspective, while the optical injection regenerates itself owing to the injection pulling effect [30], oscillation sidebands sharply emerge because of the red shifting effect [15, 31]. The oscillation sidebands are equally separated from the regeneration by an oscillation frequency which can be continuously tuned from a few gigahertz to tens or even hundreds of gigahertz by simply adjusting the injection power and frequency, thus giving rise to a characteristic of self-sustained microwave oscillation. Attributed to the red-shifted laser cavity resonance enhancement, the lower oscillation sideband typically has a power that is not only one to two orders of magnitude higher than the upper oscillation sideband but is also close to the regeneration. These unique characteristics of the P1 dynamics have attracted much research interest for both fundamental understanding [32–37] and novel applications [38–48].

The characteristic of self-sustained microwave oscillation makes the optically injected laser at the P1 dynamics inherently an all-optical microwave oscillator that is frequency-tunable continuously from a few gigahertz to tens or even hundreds of gigahertz without suffering from intrinsic laser responses and limited electronic bandwidths [19, 38, 43, 44]. The characteristic of two dominant frequency components with similar power manifests a feature of optical single-sideband modulation and also a feature of nearly 100% optical modulation depth. While the former feature is highly preferred for microwave distribution through fibers in order to mitigate microwave power fading [15, 43, 44], the latter one gives rise to maximum microwave power generation for a given optical power received by a photodetector [15, 43, 44], which improves detection sensitivity, link gain, and transmission distance. However, the spontaneous emission noise that is intrinsic to the injected laser deteriorates the spectral purity of such generated microwaves, leading to a considerably broad 3-dB microwave linewidth, typically on the order of 1 to 10 MHz, and severely poor microwave phase noise [19, 22–24]. In addition, fluctuations in the power and frequency of the optical injection relative to those of the injected laser result in significant microwave frequency jitters, typically on the order of 100 MHz [23, 36]. These characteristics limit the practical application scope of the optically injected laser at the P1 dynamics.

A few microwave stabilization schemes have therefore been proposed and demonstrated to improve the microwave spectral purity and stability of the P1 dynamics. They include locking to an electronic microwave reference through direct modulation of the injected laser [13, 26] or through optical modulation sideband injection locking [23], and locking to the P1 dynamics itself through optoelectronic feedback [14, 27] or through coherent/incoherent optical feedback [19, 20, 22, 24]. Even though the first three schemes are very effective in stabilizing the P1 microwave oscillation with a 3-dB microwave linewidth down to 1 kHz or even less than 1 Hz [13, 14, 23, 26, 27], they require certain electronic microwave devices, such as oscillators, amplifiers, and/or attenuators, which become difficult and/or expensive to implement for high-frequency microwave generation. On the contrary, the last scheme based on optical feedback bypasses the bandwidth restriction of electronics, which makes an optically injected laser at the P1 dynamics a truly all-optical microwave oscillator, and is therefore much preferred and attractive for high-frequency microwave applications. Prior research works [19, 20, 22, 24] studying the optical feedback scheme have emphasized only on a feedback delay time that is close to or longer than the relaxation resonance period of a laser at free running, which is typically on the order of 0.1 to 1 ns. This requires a feedback delay loop of a few centimeters to tens of meters, which is unfavourable for practical applications where system miniaturization is, in general, highly preferred.

With the technology advance in photonic integrated circuits, a feedback delay loop as short as a few millimeters or even a few hundreds of micrometers has been made feasible [49–51], and has therefore been much more attractive for practical applications [52, 53]. Such an ultrashort feedback delay loop corresponds to a feedback delay time that is one to two orders of magnitude shorter than the relaxation resonance period of a typical laser. As reported in the literature where a semiconductor laser subject to optical feedback only was investigated [49–51], a short feedback delay time excites various dynamical characteristics that are considerably different from those induced by a long feedback delay time. Therefore, if instead a feedback delay time that is one or two orders of magnitude shorter than the relaxation resonance period of a laser is used, it raises interesting yet fundamental questions of how an optically injected laser at the P1 dynamics reacts to such ultrashort optical feedback and, more importantly, to the interest of this study, whether the optical feedback scheme still works for microwave stabilization. Hence, in this study, we numerically investigate how ultrashort optical feedback, ranging from one to two orders of magnitude shorter than the relaxation resonance period of a laser under study, affects the dynamical and microwave characteristics of an optically injected semiconductor laser at the P1 dynamics under various operating conditions. Following this introduction, a simulation model is presented in Section 2. Results and analyses are reported in Section 3. Discussions are conducted in Section 4. Finally, a conclusion is made in Section 5.

## 2. Numerical Model

The photonic microwave generation system considered in this study basically consists of a single-mode semiconductor laser that is subject to both unidirectional continuous-wave optical injection from another laser and single-pass optical feedback from its own output. For the interest of this study, the injection condition is chosen so as to excite the P1 dynamics of the laser when subject to optical injection only. The laser system can be modeled by the following normalized rate equations [22, 24], which are equivalent to the well-known Lang-Kobayashi equations:

*a*and $\tilde{n}$ are the normalized field amplitude and carrier density of the laser, respectively, while

*ϕ*is the phase difference between the optical injection signal and the laser. Laser intrinsic parameters,

*γ*

_{c},

*γ*

_{s},

*γ*

_{n},

*γ*

_{p}, and

*b*are the cavity decay rate, spontaneous carrier relaxation rate, differential carrier relaxation rate, nonlinear carrier relaxation rate, and linewidth enhancement factor, respectively. The normalized Langevin noise-source parameters

*F*and

_{a}*F*are characterized by a spontaneous emission rate

_{ϕ}*R*

_{sp}[28]. The normalized bias current, $\tilde{J}$, represents the bias level above the threshold of the laser. The normalized injection parameter,

*ξ*

_{i}, is proportional to the ratio of the optical fields between the optical injection signal and the laser at free running, the square of which is proportional to the injection power actually received by the laser. The detuning frequency,

*f*

_{i}= Ω

_{i}/2

*π*, is the frequency offset of the optical injection signal from the free-running frequency of the laser. The normalized feedback parameter,

*ξ*

_{f}, measures the strength of the optical feedback signal. The feedback delay time,

*τ*

_{f}, is the time required for the optical feedback signal travelling back to the laser. The phase factor,

*θ*, is the phase difference between the optical feedback field and the intracavity field at the feedback injection point. Throughout the numerical calculation,

*θ*is set equal to zero to simplify our study except when it becomes the subject of interest for analysis and discussion.

The values of the intrinsic parameters adopted in this study are *γ*_{c} = 5.36 × 10^{11} s^{−1}, *γ*_{s} = 5.96 × 10^{9} s^{−1}, *γ*_{p} = 1.91 × 10^{10} s^{−1}, *γ*_{n} = 7.53 × 10^{9} s^{−1}, and *b* = 3. These values were experimentally determined [9] when the laser under consideration was biased at 40 mA, corresponding to $\tilde{J}=1.222$, with an output power of 4.5 mW. The relaxation resonance frequency of the free-running laser is given by *f*_{r} = (2*π*)^{−1}(*γ*_{c}*γ*_{n} + *γ*_{s}*γ*_{p})^{1/2} ≈ 10.25 GHz, which leads to a relaxation resonance period, *τ*_{r}, of about 98 ps. The spontaneous emission rate is taken to be *R*_{sp} = 4.7 × 10^{19} V^{2}m^{−1}s^{−1}. A second-order Runge-Kutta method with the measured laser parameters is used to solve Eqs. (1)–(3), which has been demonstrated to reproduce experimentally observed phenomena in a laser subject to, for example, optical injection only [28, 29] or both optical injection and optical feedback [22, 24]. For the purpose of this study, a time duration of about 0.47 ps is used for one integration step throughout the investigation so that *τ*_{f} that is two orders of magnitude shorter than *τ*_{r} can be considered. For one complete integration, while a time duration of 0.25 ms is adopted for the linewidth analysis of generated microwaves, which gives rise to a frequency resolution of about 4 kHz, a time duration of 1 *µ*s is used for the rest of the other analyses in order to relax the strict requirement on the computation capability and time of computers.

## 3. Results and Analyses

#### 3.1. Changes to P1 dynamical characteristics

Representative optical and microwave spectra are shown in Fig. 1 to demonstrate how the optically injected semiconductor laser operating at the P1 dynamics responds to its own optical feedback at *τ*_{f} that is one to two orders of magnitude shorter than *τ*_{r} = 98 ps. While the laser intrinsic noise is not considered in the numerical calculation for the red curves shown in Fig. 1 in order to identify the resulting dynamical states through their inherent characteristics, it is taken into account for the gray curves so as to analyze the linewidth and phase noise of the resulting microwaves. The *x* axes of the optical spectra in Fig. 1 are relative to the free-running frequency of the laser.

Let us first consider the situation when the laser is only subject to optical injection at (*ξ*_{i}, *f*_{i}) = (0.4, 40 GHz), as presented in Fig. 1(a–i), where a typical P1 dynamical state is excited. A regeneration of the optical injection itself emerges at the offset frequency of 40 GHz, which results from the injection pulling effect [30]. In addition, oscillation sidebands, which are equally separated from the regeneration by an oscillation frequency *f*_{0} = 46.06 GHz, sharply appear through undamping the relaxation resonance of the laser [29] owing to the red shifting effect [15, 31]. Attributed to the laser cavity resonance red-shift induced by the optical injection, the lower oscillation sideband is resonantly enhanced as opposed to the upper one, and therefore has a power only 4 dB weaker than the regeneration. This leads to the dominance of the regeneration and the lower oscillation sideband, thus making the optically injected laser at the P1 dynamical state effectively a two-tone optical oscillator, and this also results in a feature of optical single-sideband modulation with nearly 100% optical modulation depth. These key features of the P1 dynamical state are similarly observed when the laser noise is considered except the appearance of spectral broadening and a noise pedestal. After photodetection of the P1 dynamical state, as shown in Fig. 1(a–ii), a microwave signal at *f*_{0} = 46.06 GHz is generated with a 3-dB linewidth, Δ*ν*, of 90.7 MHz when the laser noise is considered. Note that, in this study, microwave linewidth is estimated by measuring the 3-dB linewidth of the Lorentzian fitting curve of each microwave spectrum, shown as the inset of each plot in Fig. 1.

Let us now send a fraction of the optical signal presented in Fig. 1(a–i) back to the laser itself at *τ*_{f} that is one to two orders of magnitude shorter than *τ*_{r}. As shown in Fig. 1(b–i) where (*ξ*_{f}, *τ*_{f}) = (0.1, 0.95 ps), key P1 features are similarly observed no matter whether the laser noise is considered or not, suggesting that the resulting dynamical state is intrinsically a P1 state. There exist, however, modifications to the P1 features after the ultrashort optical feedback is introduced. While the regeneration of the optical injection still appears at the offset frequency of 40 GHz, the oscillation sidebands shift considerably away from the regeneration by about 9.23 GHz, resulting in a significantly enhanced *f*_{0} = 65.29 GHz. This *f*_{0} enhancement becomes more significant when *ξ*_{f} is further increased under the same *τ*_{f}. For example, as shown in Fig. 1(c–i) where (*ξ*_{f}, *τ*_{f}) = (0.4, 0.95 ps), while similar P1 features are still observed, *f*_{0} increases more considerably to 116.77 GHz. The significant modification in *f*_{0} can be more easily identified in Figs. 1(b–ii) and 1(c–ii), where considerably frequency-shifted microwaves at *f*_{0} = 65.29 and 116.77 GHz are generated, respectively. In addition, the microwave linewidth Δ*ν* is observed to reduce from 90.7 MHz down to 57.5 and 9.5 MHz, respectively. Similar to those shown in Fig. 1(a–ii), there are harmonics beyond the frequency range considered in Figs. 1(b–ii) and 1(c–ii), which are more than 30 dB weaker than the fundamental signals at *f*_{0} = 65.29 and 116.77 GHz, respectively.

The *f*_{0} enhancement mentioned above becomes less significant if instead *τ*_{f} is further increased under the same *ξ*_{f}. For example, as shown in Figs. 1(d–i) and 1(d–ii) where (*ξ*_{f}, *τ*_{f}) = (0.1, 23.5 ps), while key P1 features are similarly observed, *f*_{0} increases only slightly to 47.4 GHz from 46.06 GHz presented in Figs. 1(a–i) and 1(a–ii). In addition, small pumps appear midway between the spectral components when the laser noise is considered, which is more easily identified in Fig. 1(d–ii), suggesting the onset of a period-doubling bifurcation from a P1 dynamical state to a period-two dynamical state. Indeed, a continuous enhancement of *τ*_{f} leads to well-defined period-two dynamical states followed by quasi-periodic dynamical states. As also presented in Fig. 1(d–ii), the microwave linewidth Δ*ν* is found to reduce substantially down to 3.22 MHz from 90.7 MHz presented in Fig. 1(a–ii). Comparing Fig. 1(d–ii) with Fig. 1(b–ii) indicates that a longer *τ*_{f} is more effective in narrowing the linewidth of the generated microwaves. Moreover, the comparison between Figs. 1(d–ii) and 1(c–ii) suggests that a higher *ξ*_{f} helps to reduce the microwave linewidth to a similar level if a shorter *τ*_{f} is used instead.

Similar characteristic modifications, yet with different extents, of the P1 dynamical state and similar excitation of various other dynamical states by suppressing the P1 dynamical state are observed over a variety of different ultrashort optical feedback conditions. Figure 2 summarizes the observed dynamical states as a function of *ξ*_{f} and *τ*_{f} under study. Each different dynamical state is identified by the different number of its intensity extremum obtained from the time series of the laser output when the laser noise is not considered in the numerical calculation. Generally speaking, P1 dynamics (red regions) exhibit two distinct intensity extrema, period-two dynamics (blue regions) show four distinct intensity extrema, quasi-periodic dynamics (gray regions) reveal two groups of continuously distributed intensity extrema with a gap in between, and chaos and other instabilities (black regions) demonstrate continuously distributed intensity extrema. While a continuous distribution of the P1 dynamics is observed around the lower left corner of Fig. 2, they also appear strip-wise along the diagonal direction with the quasi-periodic dynamics and chaos emerging in between.

#### 3.2. Changes to P1 microwave frequency

Changes to the microwave characteristics of the P1 dynamics are evidently observed in Sec. 3.1 when the ultrashort optical feedback is introduced to the optically injected laser. For further understanding, let us first investigate how the microwave frequency actually varies under different ultrashort optical feedback conditions, as presented in Figs. 3 and 7. To study the intrinsic variation of the microwave frequency, the laser noise is not considered in the numerical calculation.

### 3.2.1. Effect of optical feedback strength

Figure 3(a) shows that *f*_{0} generally enhances as *ξ*_{f} is increased yet with a different manner for a different *τ*_{f}. For *τ*_{f} = 0.95 ps, shown as the black squares, *f*_{0} increases monotonically and continuously with *ξ*_{f} and enhances up to 116.77 GHz at *ξ*_{f} = 0.4 under study, an enhancement of nearly three folds from 46.06 GHz when no optical feedback is introduced. Such a continuous *f*_{0} enhancement as a function of *ξ*_{f} is similarly observed for *τ*_{f} up to around 9.5 ps, beyond which dynamical states other than the P1 dynamics appear, as clearly observed in Fig. 2. However, the level of the *f*_{0} enhancement is found to reduce with *τ*_{f}. For example, for *τ*_{f} = 9.5 ps, shown as the red circles, *f*_{0} also increases continuously with *ξ*_{f} but only enhances up to about 58 GHz at *ξ*_{f} = 0.4. On the other hand, for *τ*_{f} = 23.5 ps, shown as the blue up-triangles, *f*_{0} enhances step-wise with *ξ*_{f} and increases up to 126.1 GHz at *ξ*_{f} = 0.4. As referred to Fig. 2, each step-like *f*_{0} enhancement is found to occur when the laser system re-enters the P1 dynamics from other dynamical states. Such a step-like *f*_{0} enhancement in terms of *ξ*_{f} happens for *τ*_{f} > 9.5 ps under study. For example, for *τ*_{f} = 30.51 ps, shown as the green down-triangles, *f*_{0} also increases step-wise with *ξ*_{f} and enhances up to 138.8 GHz at *ξ*_{f} = 0.4.

The considerable *f*_{0} enhancement with *ξ*_{f}, no matter whether continuously or step-wise, can be qualitatively understood from the viewpoint of the laser cavity resonance shift. The introduction of an external optical field, either optical injection or optical feedback, reduces the necessary gain for the laser from its free-running value [15, 31]. This results in the increase of the refractive index of the laser cavity through the linewidth enhancement factor *b*, which is commonly known as the antiguidance effect. Consequently, the laser cavity resonance is shifted from its free-running value by [15, 31]

By introducing the optical feedback to the optically injected laser, the laser cavity resonance shifts further away from its free-running value and, accordingly, the lower oscillation sideband shifts further away from the regeneration of the optical injection. In general, the stronger the optical feedback is, the more the laser cavity resonance and therefore the lower oscillation sideband shift. This is demonstrated in Fig. 3(b) as a function of *ξ*_{f} at, for example, *τ*_{f} = 0.95 and 23.5 ps, respectively. The lower oscillation sideband frequency relative to the free-running laser oscillation is obtained with Eqs. (1)–(3), as the closed symbols show, and the shifted laser cavity resonance frequency, *ω*_{s}/2*π*, is calculated using Eq. (4), as the open symbols present. As observed, the lower oscillation sideband indeed emerges around the red-shifted laser cavity resonance, about 8 to 13 GHz away under study, and red-shifts more as *ξ*_{f} is increased. Accordingly, the considerable *f*_{0} enhancement with *ξ*_{f} observed in Fig. 3(a) results from the strong red-shift of the laser cavity resonance, which is made possible for ultrashort optical feedback where adequately high *ξ*_{f} can be introduced while keeping the laser system at the P1 dynamics. Note that the frequency deviation between the shifted laser cavity resonance and the lower oscillation sideband shown in Fig. 3(b) results from the direct effect of the optical injection and feedback. As observed in Eq. (2). While the first term calculates the indirect effect of both external perturbations through the laser cavity resonance shift, the second and third terms take into account their direct effects, all of which are required to determine the emergence of the P1 oscillation sidebands.

### 3.2.2. Effect of optical feedback delay time

The result shown in Fig. 3(a) suggests that the *f*_{0} enhancement depends on *τ*_{f} as well, which can be more easily observed in Fig. 4(a). While *f*_{0} enhances considerably from 46.06 GHz after the ultrashort optical feedback is introduced, it reduces continuously with *τ*_{f}, in general, but enhances abruptly at certain *τ*_{f} values for all four different *ξ*_{f} values under study. On one hand, the considerable enhancement of *f*_{0} can again be qualitatively understood from the perspective of the strong red-shift in the laser cavity resonance after the ultrashort optical feedback is introduced, as presented in Fig. 4(b) in terms of *τ*_{f} at, for example, *ξ*_{f} = 0.1 and 0.4, respectively. On the other hand, the saw-like variation in *f*_{0} as a function *τ*_{f}, which is also similarly observed for long optical feedback [22], is strongly related to the behavior of locking between the microwave oscillation of the P1 dynamics and the resonance modes of the optical feedback loop, as demonstrated in the following analysis and discussion.

When the optical feedback is introduced, the optically injected laser is required to satisfy an extra resonance condition formed by the feedback loop and, as a result, the microwave oscillation of the P1 dynamics is forced to lock on to the closest feedback loop mode. A gradual enhancement of *τ*_{f} leads to a gradual shift of every feedback loop mode toward a lower frequency. This therefore results in a gradual reduction of *f*_{0} until *τ*_{f} reaches a value when two feedback loop modes appear approximately equally from the microwave oscillation of the P1 dynamics. A slight further enhancement of *τ*_{f} would lock the P1 dynamics to the feedback loop mode at a higher frequency, giving rise to an abrupt *f*_{0} enhancement, Δ*f*_{0}, equal to the frequency separation between the two feedback loop modes, i.e., the feedback loop frequency *f _{l}* = 1/

*τ*

_{f}. The same process repeats itself with a period

*T*

_{1}that is equal to the reciprocal of the frequency difference between the lower oscillation sideband of the modified P1 dynamics under ultrashort optical feedback and the free-running laser oscillation.

The analysis presented in Fig. 5(a) reveals that each Δ*f*_{0} shown in Fig. 4(a) is close to each corresponding *f _{l}*, i.e., the reciprocal of each corresponding

*τ*

_{f}value around which the abrupt

*f*

_{0}enhancement occurs. Note that, by referring to Fig. 2, each abrupt

*f*

_{0}enhancement is observed to happen when the laser system re-enters the P1 dynamics from other dynamical states. This indicates that the effect of feedback loop mode competition becomes considerably strong around each

*τ*

_{f}value leading to the abrupt

*f*

_{0}enhancement, where two feedback loop modes appear approximately equally from the microwave oscillation of the P1 dynamics and each of the two feedback loop modes attempts to lock the microwave oscillation of the P1 dynamics on to itself. Such strong mode competition therefore results in the excitation of dynamical states that are more complicated than the P1 dynamics, such as the quasi-periodic and chaotic dynamics, over a range of

*τ*

_{f}around each abrupt

*f*

_{0}enhancement. The appearance of other dynamical states over a range of

*τ*

_{f}makes it difficult to identify the exact value of each

*τ*

_{f}leading to each abrupt

*f*

_{0}enhancement and to estimate the actual Δ

*f*

_{0}of each abrupt

*f*

_{0}enhancement, both of which lead to the deviation between Δ

*f*

_{0}and

*f*shown in Fig. 5. Nevertheless, the behaviors of Δ

_{l}*f*

_{0}and

*f*as a function of

_{l}*τ*

_{f}are qualitatively similar. The analysis presented in Fig. 5(b) demonstrates that

*T*

_{1}shown in Fig. 4(a) is indeed equal to the reciprocal of the frequency difference between the lower oscillation sideband of the modified P1 dynamics under ultrashort optical feedback and the free-running laser oscillation. Since the lower oscillation sideband of the modified P1 dynamics generally shifts away from the free-running laser oscillation as

*ξ*

_{f}is increased, as addressed in Fig. 4(a),

*T*

_{1}becomes shorter for a stronger

*ξ*

_{f}.

The mapping shown in Fig. 6 summarizes how *f*_{0} depends on both *ξ*_{f} and *τ*_{f} from a global viewpoint. As clearly observed, despite from the emergence of dynamical states other than the P1 dynamics, *f*_{0} can be considerably increased from 46.06 GHz after the ultrashort optical feedback is introduced, up to more than 140 GHz under study. In general, the stronger the optical feedback is, the higher *f*_{0} becomes. This provides an extra degree of freedom for the optically injected laser system to manipulate the frequency of generated microwaves without changing the optical injection condition while the purity of the generated microwaves is improved through optical feedback, as demonstrated in the following subsection.

### 3.2.3. Effect of optical feedback phase

As reported in the literature where a semiconductor laser subject to optical feedback only was investigated [49, 50], the phase factor *θ* appearing in Eqs. (1) and (2) plays a more significant role in dynamical evolution for *τ*_{f} shorter than *τ*_{r}. For the optically injected laser at the P1 dynamics with ultrashort optical feedback under study, this has stimulated us to wonder whether and how the microwave frequency changes with *θ*. Let us look into this issue by studying a mapping of *f*_{0} as a function of *θ* and *ξ*_{f} at *τ*_{f} = 23.5 ps, as shown in Fig. 7(a), to obtain a macroscopic understanding. First, dynamical states other than the P1 dynamics, such as stable locking, period-two dynamics, quasi-periodic dynamics, and chaos, shown as the white regions, emerge at certain *θ* values that depend on *ξ*_{f}. This suggests that *θ* plays a key role in determining the dynamical behavior of the laser system under study. Second, except for relatively weak *ξ*_{f}, *f*_{0} generally increases with *θ* along a fixed *ξ*_{f} within one P1 dynamical region. It, however, reduces step-wise with *θ* when the laser system enters from one P1 dynamical region to another one where other dynamical states appear in between. Over the range of (*θ*, *ξ*_{f}) under consideration, *f*_{0} generally enhances from 46.06 GHz when no optical feedback is introduced.

To look into the issue from another viewpoint, a mapping of *f*_{0} as a function of *θ* and *τ*_{f} at *ξ*_{f} = 0.4 is presented in Fig. 7(b). Similarly, dynamical states other than the P1 dynamics, also shown as the white regions, appear at specific *θ* values that yet depend on *τ*_{f}. For relatively short *τ*_{f}, *f*_{0} typically enhances with *θ* along a fixed *τ*_{f} within one P1 dynamical region. It, however, decreases step-wise with *θ* when the laser system enters from one P1 dynamical region to another one where other dynamical states appear in between. Such a varying behavior of *f*_{0} as a function of *θ* becomes less dynamic if *τ*_{f} is increased. As a result, for relatively long *τ*_{f}, *f*_{0} becomes much less sensitive to *θ*. Over the range of (*θ*, *τ*_{f}) under study, *f*_{0} generally enhances from 46.06 GHz when no optical feedback is introduced.

#### 3.3. Changes to P1 microwave purity

Let us now study how the microwave spectral purity of the P1 dynamics, namely linewidth and phase noise, changes under different ultrashort optical feedback conditions, as presented in Figs. 8 and 11 where the laser noise is taken into account in the numerical calculation in order to analyze these microwave phase characteristics. To quantify the phase noise over a broad range, the phase noise variance is estimated by integrating the single-sideband phase noise of the resulting microwaves from the frequency offset of 1 MHz to 500 MHz [19, 22]. For the range of *τ*_{f} under study, this integration does not include feedback loop modes, thus eliminating the contribution of the phase noise from the modes.

### 3.3.1. Effect of optical feedback strength

Figures 8(a) and 8(b) demonstrate the 3-dB microwave linewidth Δ*ν* and the phase noise variance in terms of *ξ*_{f} at *τ*_{f} = 0.95 and 23.5 ps, respectively, when (*ξ*_{i}, *f*_{i}) = (0.4, 40 GHz). As observed, no matter whether other dynamical states appear or not, as referred to Fig. 3(a), both microwave phase characteristics of the P1 dynamics generally reduce while *ξ*_{f} is increased. A reduction of up to three orders of magnitude in either Δ*ν* or the phase noise can be achieved as compared with the result when no optical feedback is introduced. This suggests that the P1 dynamics is locked on to its own optical feedback more strongly if a higher *ξ*_{f} is used.

### 3.3.2. Effect of optical feedback delay time

The results presented in Fig. 8 also suggest that the improvement of the microwave spectral purity also depends on *τ*_{f}. This can be more easily observed in Fig. 9, where Δ*ν* and the phase noise variance in terms of *τ*_{f} at *ξ*_{f} =0.1 and 0.4, respectively, when (*ξ*_{i}, *f*_{i}) = (0.4, 40 GHz) are presented. Globally speaking, no matter whether other dynamical states emerge or not, as referred to Fig. 4(a), both microwave characteristics of the P1 dynamics decrease as *τ*_{f} is increased. A reduction of more than three orders of magnitude in either Δ*ν* or the phase noise can be achieved as compared with the result when no optical feedback is introduced. This suggests that the P1 dynamics is locked on to its own optical feedback more strongly if a longer *τ*_{f} is used as the optical feedback loses more its coherence with the intracavity optical field of the laser. Locally speaking, however, Δ*ν* and the phase noise variance enhance at certain *τ*_{f} values where the laser system is about to enter dynamical states other than the P1 dynamics or, equivalently, where an abrupt *f*_{0} enhancement is about to happen, as referred to Fig. 4(a). By comparing Figs. 9(a) and 9(b), the enhancement of either Δ*ν* or the phase noise variance becomes much less significant as *ξ*_{f} is increased.

The mapping shown in Fig. 10 summarizes how the phase noise variance depends on both *ξ*_{f} and *τ*_{f} from a global viewpoint. As observed, after the ultrashort optical feedback is introduced, the phase noise variance can be reduced significantly by up to more than three orders of magnitude from 28.17 rad^{2} under study. In general, the stronger or longer the optical feedback is, the lower the phase noise variance becomes.

### 3.3.3. Effect of optical feedback phase

To understand the effect of *θ* on the microwave spectral purity of the P1 dynamics, Fig. 11(a) presents a mapping of the phase noise variance as a function of *θ* and *ξ*_{f} at *τ*_{f} = 23.5 ps. Except for the boundaries that separate P1 dynamical regions from other dynamical regions, the phase noise variance changes slightly with *θ*, typically less than an order of magnitude, along a fixed *ξ*_{f} within a P1 dynamical region. While the phase noise variance generally reduces over the range of (*θ*, *ξ*_{f}) under consideration from the value of 28.17 rad^{2} when no optical feedback is introduced, it enhances up to more than an order of magnitude at certain *θ* values that depend on *ξ*_{f}, such as those around *θ* = *π* where a stable locking region appears in this example of demonstration. On the other hand, a mapping of the phase noise variance as a function of *θ* and *τ*_{f} at *ξ*_{f} = 0.4 is presented in Fig. 11(b). Similarly, except for the boundaries that separate P1 dynamical regions from other dynamical regions, the phase noise variance varies slightly with *θ*, generally less than an order of magnitude, along a fixed *τ*_{f} within a P1 dynamical region. While the phase noise variance generally decreases over the range of (*θ*, *τ*_{f}) under study from the value of 28.17 rad^{2} when no optical feedback is introduced, it enhances up to more than an order of magnitude at certain *θ* values that depend on *τ*_{f}, such as those around *θ* = *π* where a stable locking region appears in this example of demonstration. The results shown in Fig. 11 suggest that, while the ultrashort optical feedback is effective in improving the microwave spectral purity of the P1 dynamics, the improvement is much less dependent on *θ* as compared with *ξ*_{f} and *τ*_{f}.

## 4. Discussion

Before concluding this study, a few remarks are made as follows to compare the effects of ultrashort optical feedback demonstrated here with those of long optical feedback investigated in [19, 20, 22, 24] on microwave generation using the optically injected laser at the P1 dynamics. First, compared with the case of long optical feedback [22], there exist many more optical feedback conditions leading to the P1 dynamics for ultrashort optical feedback, particularly at high *ξ*_{f}. This provides the optically injected laser system with an extra degree of freedom to manipulate the microwave characteristics of the P1 dynamics without changing the optical injection condition for practical applications.

Second, while *f*_{0} enhances significantly from 46.06 GHz up to more than 140 GHz for ultrashort optical feedback, it only varies slightly within a few gigahertz for long optical feedback. The considerable *f*_{0} enhancement results from the strong red-shift of the laser cavity resonance, which is made possible for ultrashort optical feedback where adequately high *ξ*_{f} can be introduced while keeping the laser system at the P1 dynamics. By contrast, for long optical feedback, the highest *ξ*_{f} that can be introduced is generally an order of magnitude lower, beyond which the laser system enters dynamical states other than the P1 dynamics. The laser cavity resonance shift is therefore relatively weak for long optical feedback, resulting in small *f*_{0} variation.

Third, for either long or ultrashort optical feedback, a longer *τ*_{f} is more effective in improving the spectral purity of the generated microwaves. A higher *ξ*_{f} is also more effective in enhancing the microwave purity for ultrashort optical feedback. This is, however, not the case for long optical feedback as the highest *ξ*_{f} that can be introduced is not adequately high for microwave purity improvement [22, 24]. Consequently, *ξ*_{f} plays a more effective role in improving the spectral purity of the generated microwaves at the regime of ultrashort optical feedback. Hence, even though *τ*_{f} is much shorter for ultrashort optical feedback than long optical feedback, the capability of increasing *ξ*_{f} to adequately high levels for the former helps to improve the microwave purity to an extent similar to or even better than that for the latter.

Fourth, owing to the resonance modes formed by an optical feedback loop, side peaks appear around the P1 microwave oscillation by approximately *f _{l}* = 1/

*τ*

_{f}[19, 20, 22, 24]. For long optical feedback where

*τ*

_{f}is on the order of 1 to 100 ns, these side peaks are only a few megahertz to a few gegahertz away from the P1 microwave oscillation. This not only give rises to degradation of the phase noise in the generated microwaves [19, 20, 22, 24], but also results in laser-noise-induced frequency jitter of the P1 microwave oscillation due to mode hopping [22]. A second optical feedback is therefore required to suppress the side peaks in order to improve the microwave purity and to prevent the frequency jitter [19, 24]. By contrast, for ultrashort optical feedback where

*τ*

_{f}is on the order of 1 to 100 ps, since the side peaks are at least a few tens of gigahertz away from the P1 microwave oscillation, no degradation of the microwave phase noise and no frequency jitter of the P1 microwave oscillation happen. Therefore, there is no need of a second optical feedback loop, which shall simplify the structure of the photonic microwave generation system.

For the purpose of practical applications where system miniaturization is generally preferred, a feedback delay time that is one to two orders of magnitude shorter than the relaxation resonance period of a laser is emphasized in this study. This corresponds to a feedback delay loop of a few hundreds of micrometers to a few millimeters, which is made feasible with the technology advance in photonic integrated circuits. For example, based on the works done in [50, 51], a monolithically integrated optical feedback system can be practically fabricated, which comprises a 300 *µ*m laser section followed by a 100 *µ*m gain/absorption section, a 150 *µ*m phase section, and a high-reflection coating at the end of the phase section. The optical feedback loop is formed by the gain/absorption section, the phase section, and the coating, which corresponds to *τ*_{f} of about 6 ps in this design. The optical feedback power and phase can be adjusted through manipulating the bias current of the gain/absorption and phase sections, respectively. According to the work of [50], *θ* can be simply adjusted over 2*π* for every 6.2 mA using a 150 *µ*m phase section. The length of either the gain/absorption section or the phase section can be further decreased or increased to acquire a different *τ*_{f} for practical applications of this study. The bias current of either section needs to be adjusted accordingly to achieve a similar level of *ξ*_{f} or *θ*. A passive waveguide with a length of a few hundreds of micrometers to a few millimeters can also be integrated into the system if a much longer *τ*_{f}, such as tens of ps, is required.

The capability to manipulate or improve the characteristics of the generated microwaves using an optically injected semiconductor laser at the P1 dynamics through ultrashort optical feedback is highly beneficial in practical applications. For example, for a typical situation where the feedback delay time is fixed, the purity of the P1 microwave oscillation can be improved by up to more than three orders of magnitude through increasing the optical feedback power. The enhancement of its microwave frequency that follows with the increment of the optical feedback power can be compensated by adjusting the optical feedback phase, which only slightly changes the improved microwave purity, so as to keep its microwave frequency unchanged. As another example, under a fixed feedback power level and a fixed feedback delay time, the frequency of the P1 microwave oscillation can be adjusted within a range of a few tens of gigahertz through changing the optical feedback phase while keeping the microwave purity at a similar level.

## 5. Conclusion

This study numerically investigates the enhancement of photonic microwave generation using an optically injected semiconductor laser operating at the P1 dynamics through ultrashort optical feedback. Various dynamical states that are more complicated than the P1 dynamics can be excited under a variety of different ultrashort optical feedback conditions. Within the range of the P1 dynamics, on one hand, the frequency of the P1 microwave oscillation can be greatly enhanced by up to about three folds, which is attributed to the strong red-shift of the laser cavity resonance. In general, the stronger the ultrashort optical feedback is, the more the microwave frequency is enhanced. In addition, the frequency of the P1 microwave oscillation varies saw-wise with the feedback delay time, which results from the behavior of locking between the P1 microwave oscillation and the optical feedback loop modes. Moreover, the frequency of the P1 microwave oscillation generally enhances with the optical feedback phase. On the other hand, the purity of the P1 microwave oscillation can be highly improved by up to more than three orders of magnitude. The stronger or longer the ultrashort optical feedback is, the more the microwave purity is improved. Except for the boundaries that separate the P1 dynamics from other dynamical states, the microwave purity only varies within an order of magnitude with the optical feedback phase. These results suggest that the ultrashort optical feedback provides the optically injected laser system with an extra degree of freedom to manipulate/improve the characteristics of the P1 microwave oscillation without changing the optical injection condition.

## Funding

Ministry of Science and Technology of Taiwan (MOST103-2112-M-006-013-MY3, MOST106-2112-M-006-004-MY3).

## Acknowledgments

The authors would like to thank Professor Shih-Hui Chang at the Department of Photonics, National Cheng Kung University, Taiwan for kindly providing them with a computer cluster for the numerical calculation of microwave linewidth.

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