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:

dadt=12[γcγnγsJ˜n˜γp(2a+a2)](1+a)+ξiγccos(Ωit+ϕ)+ξfγc[1+a(tτf)]cos[ϕ(tτf)ϕ(t)+θ]+Fa
dϕdt=b2[γcγnγsJ˜n˜γp(2a+a2)]ξiγc1+asin(Ωit+ϕ)+ξfγc1+a(tτf)1+a(t)sin[ϕ(tτf)ϕ(t)+θ]+Fϕ1+a
dn˜dt=γsn˜γn(1+a)2n˜γsJ˜(2a+a2)+γsγpγcJ˜(2a+a2)(1+a)2
Here, a and 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, γc, γs, γn, γp, 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 Fa and Fϕ are characterized by a spontaneous emission rate Rsp [28]. The normalized bias current, J˜, represents the bias level above the threshold of the laser. The normalized injection parameter, ξi, 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, fi = Ωi/2π, is the frequency offset of the optical injection signal from the free-running frequency of the laser. The normalized feedback parameter, ξf, measures the strength of the optical feedback signal. The feedback delay time, τf, 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 γc = 5.36 × 1011 s−1, γs = 5.96 × 109 s−1, γp = 1.91 × 1010 s−1, γn = 7.53 × 109 s−1, and b = 3. These values were experimentally determined [9] when the laser under consideration was biased at 40 mA, corresponding to J˜=1.222, with an output power of 4.5 mW. The relaxation resonance frequency of the free-running laser is given by fr = (2π)−1(γcγn + γsγp)1/2 ≈ 10.25 GHz, which leads to a relaxation resonance period, τr, of about 98 ps. The spontaneous emission rate is taken to be Rsp = 4.7 × 1019 V2m−1s−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 τf that is two orders of magnitude shorter than τr 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 µs 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 τf that is one to two orders of magnitude shorter than τr = 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.

 figure: Fig. 1

Fig. 1 Optical spectra (left column) and microwave spectra (right column) of the laser subject to both optical injection at (ξi, fi) = (0.4, 40 GHz) and optical feedback at (ξf, τf) = (0, 0 ps) for (a–i)(a–ii), (0.1, 0.95 ps) for (b–i)(b–ii), (0.4, 0.95 ps) for (c–i)(c–ii), and (0.1, 23.5 ps) for (d–i)(d–ii), respectively. Red curves, no laser noise is considered; Gray curves, laser noise is considered. The x axes of the optical spectra are relative to the free-running frequency of the laser. Each inset is an enlargement of each corresponding microwave spectrum in a linear scale with a Lorentzian fitting curve in blue.

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Let us first consider the situation when the laser is only subject to optical injection at (ξi, fi) = (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 f0 = 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 f0 = 46.06 GHz is generated with a 3-dB linewidth, Δν, 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 τf that is one to two orders of magnitude shorter than τr. As shown in Fig. 1(b–i) where (ξf, τf) = (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 f0 = 65.29 GHz. This f0 enhancement becomes more significant when ξf is further increased under the same τf. For example, as shown in Fig. 1(c–i) where (ξf, τf) = (0.4, 0.95 ps), while similar P1 features are still observed, f0 increases more considerably to 116.77 GHz. The significant modification in f0 can be more easily identified in Figs. 1(b–ii) and 1(c–ii), where considerably frequency-shifted microwaves at f0 = 65.29 and 116.77 GHz are generated, respectively. In addition, the microwave linewidth Δν 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 f0 = 65.29 and 116.77 GHz, respectively.

The f0 enhancement mentioned above becomes less significant if instead τf is further increased under the same ξf. For example, as shown in Figs. 1(d–i) and 1(d–ii) where (ξf, τf) = (0.1, 23.5 ps), while key P1 features are similarly observed, f0 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 τf 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 Δν 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 τf 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 ξf helps to reduce the microwave linewidth to a similar level if a shorter τf 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 ξf and τf 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.

 figure: Fig. 2

Fig. 2 Dynamical mapping of the laser subject to both optical injection and optical feedback in terms of ξf and τf at (ξi, fi) = (0.4, 40 GHz). Red regions, period-one dynamics (P1); Blue regions, period-two dynamics (P2); Gray regions, quasi-periodic dynamics (QP); Black regions, chaos and other instabilities (C).

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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.

 figure: Fig. 3

Fig. 3 (a) Microwave frequency f0 in terms of ξf at τf = 0.95 ps (black squares), 9.5 ps (red circles), 23.5 ps (blue up-triangles), and 30.51 ps (green down-triangles), respectively. (b) Shifted laser cavity resonance frequency (open symbols) and lower oscillation sideband frequency (closed symbols) in terms of ξf at τf = 0.95 ps (black squares) and 23.5 ps (blue up-triangles), respectively. The y axes of (b) are relative to the free-running frequency of the laser. The injection condition is kept at (ξi, fi) = (0.4, 40 GHz).

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3.2.1. Effect of optical feedback strength

Figure 3(a) shows that f0 generally enhances as ξf is increased yet with a different manner for a different τf. For τf = 0.95 ps, shown as the black squares, f0 increases monotonically and continuously with ξf and enhances up to 116.77 GHz at ξf = 0.4 under study, an enhancement of nearly three folds from 46.06 GHz when no optical feedback is introduced. Such a continuous f0 enhancement as a function of ξf is similarly observed for τf 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 f0 enhancement is found to reduce with τf. For example, for τf = 9.5 ps, shown as the red circles, f0 also increases continuously with ξf but only enhances up to about 58 GHz at ξf = 0.4. On the other hand, for τf = 23.5 ps, shown as the blue up-triangles, f0 enhances step-wise with ξf and increases up to 126.1 GHz at ξf = 0.4. As referred to Fig. 2, each step-like f0 enhancement is found to occur when the laser system re-enters the P1 dynamics from other dynamical states. Such a step-like f0 enhancement in terms of ξf happens for τf > 9.5 ps under study. For example, for τf = 30.51 ps, shown as the green down-triangles, f0 also increases step-wise with ξf and enhances up to 138.8 GHz at ξf = 0.4.

The considerable f0 enhancement with ξf, 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]

ωs=b2[γcγnγsJ˜n˜γp(2a+a2)]
which is generally smaller than zero and thus leads to the so-called red shifting effect. This effect is involved in the phase dynamics of the laser system, as described by the first term in Eq. (2), and attempts to shift the intracavity field oscillation toward the shifted laser cavity resonance. This suggests that the red shifting effect would prompt the excitation of signals around the shifted laser cavity resonance under proper operating conditions, therefore giving rise to the modification of the laser dynamics. For the optically injected laser operating at the P1 dynamics without optical feedback, as shown in Fig. 1(a–i), such signal excitation can be clearly identified by both the emergence of the lower oscillation sideband around the shifted laser cavity resonance [15,31] and the dominance of the lower oscillation sideband over the upper one owing to the shifted laser cavity resonance enhancement.

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 ξf at, for example, τf = 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, ωs/2π, 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 ξf is increased. Accordingly, the considerable f0 enhancement with ξf 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 ξf 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 f0 enhancement depends on τf as well, which can be more easily observed in Fig. 4(a). While f0 enhances considerably from 46.06 GHz after the ultrashort optical feedback is introduced, it reduces continuously with τf, in general, but enhances abruptly at certain τf values for all four different ξf values under study. On one hand, the considerable enhancement of f0 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 τf at, for example, ξf = 0.1 and 0.4, respectively. On the other hand, the saw-like variation in f0 as a function τf, 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.

 figure: Fig. 4

Fig. 4 (a) Microwave frequency f0 in terms of τf at ξf = 0.1 (black squares), 0.2 (red circles), 0.3 (blue up-triangles), and 0.4 (green down-triangles), respectively. (b) Shifted laser cavity resonance frequency (open symbols) and lower oscillation sideband frequency (closed symbols) in terms of τf at ξf = 0.1 (black squares) and 0.4 (green down-triangles), respectively. The y axes of (b) are relative to the free-running frequency of the laser. The injection condition is kept at (ξi, fi) = (0.4, 40 GHz).

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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 τf leads to a gradual shift of every feedback loop mode toward a lower frequency. This therefore results in a gradual reduction of f0 until τf reaches a value when two feedback loop modes appear approximately equally from the microwave oscillation of the P1 dynamics. A slight further enhancement of τf would lock the P1 dynamics to the feedback loop mode at a higher frequency, giving rise to an abrupt f0 enhancement, Δf0, equal to the frequency separation between the two feedback loop modes, i.e., the feedback loop frequency fl = 1/τf. The same process repeats itself with a period T1 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 Δf0 shown in Fig. 4(a) is close to each corresponding fl, i.e., the reciprocal of each corresponding τf value around which the abrupt f0 enhancement occurs. Note that, by referring to Fig. 2, each abrupt f0 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 τf value leading to the abrupt f0 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 τf around each abrupt f0 enhancement. The appearance of other dynamical states over a range of τf makes it difficult to identify the exact value of each τf leading to each abrupt f0 enhancement and to estimate the actual Δf0 of each abrupt f0 enhancement, both of which lead to the deviation between Δf0 and fl shown in Fig. 5. Nevertheless, the behaviors of Δf0 and fl as a function of τf are qualitatively similar. The analysis presented in Fig. 5(b) demonstrates that T1 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 ξf is increased, as addressed in Fig. 4(a), T1 becomes shorter for a stronger ξf.

 figure: Fig. 5

Fig. 5 (a) Feedback loop frequency fl (black curve) and abrupt microwave frequency enhancement Δf0 at ξf =0.1 (black squares), 0.2 (red circles), 0.3 (blue up-triangles), and 0.4 (green down-triangles), respectively, in terms of τf. (b) Reciprocal of T1 (red squares) and frequency difference between the lower P1 oscillation sideband and the free-running laser oscillation (black curve) in terms of ξf.

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The mapping shown in Fig. 6 summarizes how f0 depends on both ξf and τf from a global viewpoint. As clearly observed, despite from the emergence of dynamical states other than the P1 dynamics, f0 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 f0 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.

 figure: Fig. 6

Fig. 6 Mappings of microwave frequency f0 in terms of ξf and τf at (ξi, fi) = (0.4, 40 GHz). Each number labeled in the figure indicates the microwave frequency, in GHz, along the boundary between two different gray-scaled regions.

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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 τf shorter than τr. 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 f0 as a function of θ and ξf at τf = 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 ξf. This suggests that θ plays a key role in determining the dynamical behavior of the laser system under study. Second, except for relatively weak ξf, f0 generally increases with θ along a fixed ξf 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, ξf) under consideration, f0 generally enhances from 46.06 GHz when no optical feedback is introduced.

 figure: Fig. 7

Fig. 7 Mappings of microwave frequency f0 (a) in terms of θ and ξf at τf = 23.5 ps and (b) in terms of θ and τf at ξf = 0.4, respectively, when (ξi, fi) = (0.4, 40 GHz). The white regions present dynamical states other than the P1 dynamics. Each number labeled in (a) indicates the microwave frequency, in GHz, along the boundary between two different gray-scaled regions.

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To look into the issue from another viewpoint, a mapping of f0 as a function of θ and τf at ξf = 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 τf. For relatively short τf, f0 typically enhances with θ along a fixed τf 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 f0 as a function of θ becomes less dynamic if τf is increased. As a result, for relatively long τf, f0 becomes much less sensitive to θ. Over the range of (θ, τf) under study, f0 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 τf under study, this integration does not include feedback loop modes, thus eliminating the contribution of the phase noise from the modes.

 figure: Fig. 8

Fig. 8 Microwave linewdith Δν (black symbols) and phase noise variance (red symbols) in terms of ξf at (a) τf = 0.95 ps and (b) τf = 23.5 ps, respectively, when (ξi, fi) = (0.4, 40 GHz).

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3.3.1. Effect of optical feedback strength

Figures 8(a) and 8(b) demonstrate the 3-dB microwave linewidth Δν and the phase noise variance in terms of ξf at τf = 0.95 and 23.5 ps, respectively, when (ξi, fi) = (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 ξf is increased. A reduction of up to three orders of magnitude in either Δν 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 ξf 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 τf. This can be more easily observed in Fig. 9, where Δν and the phase noise variance in terms of τf at ξf =0.1 and 0.4, respectively, when (ξi, fi) = (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 τf is increased. A reduction of more than three orders of magnitude in either Δν 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 τf is used as the optical feedback loses more its coherence with the intracavity optical field of the laser. Locally speaking, however, Δν and the phase noise variance enhance at certain τf values where the laser system is about to enter dynamical states other than the P1 dynamics or, equivalently, where an abrupt f0 enhancement is about to happen, as referred to Fig. 4(a). By comparing Figs. 9(a) and 9(b), the enhancement of either Δν or the phase noise variance becomes much less significant as ξf is increased.

 figure: Fig. 9

Fig. 9 Microwave linewdith Δν (black symbols) and phase noise variance (red symbols) in terms of τf at (a) ξf =0.1 and (b) ξf =0.4, respectively, when (ξi, fi) = (0.4, 40 GHz).

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The mapping shown in Fig. 10 summarizes how the phase noise variance depends on both ξf and τf 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 rad2 under study. In general, the stronger or longer the optical feedback is, the lower the phase noise variance becomes.

 figure: Fig. 10

Fig. 10 Mappings of phase noise variance in terms of ξf and τf at (ξi, fi) = (0.4, 40 GHz). The log-scaled values of the phase noise variance are presented.

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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 ξf at τf = 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 ξf within a P1 dynamical region. While the phase noise variance generally reduces over the range of (θ, ξf) under consideration from the value of 28.17 rad2 when no optical feedback is introduced, it enhances up to more than an order of magnitude at certain θ values that depend on ξf, 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 τf at ξf = 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 τf within a P1 dynamical region. While the phase noise variance generally decreases over the range of (θ, τf) under study from the value of 28.17 rad2 when no optical feedback is introduced, it enhances up to more than an order of magnitude at certain θ values that depend on τf, 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 ξf and τf.

 figure: Fig. 11

Fig. 11 Mappings of phase noise variance (a) in terms of θ and ξf at τf = 23.5 ps and (b) in terms of θ and τf at ξf = 0.4, respectively, when (ξi, fi) = (0.4, 40 GHz). The white regions present dynamical states other than the P1 dynamics. The log-scaled values of the phase noise variance are presented.

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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 ξf. 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 f0 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 f0 enhancement results from the strong red-shift of the laser cavity resonance, which is made possible for ultrashort optical feedback where adequately high ξf can be introduced while keeping the laser system at the P1 dynamics. By contrast, for long optical feedback, the highest ξf 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 f0 variation.

Third, for either long or ultrashort optical feedback, a longer τf is more effective in improving the spectral purity of the generated microwaves. A higher ξf 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 ξf that can be introduced is not adequately high for microwave purity improvement [22, 24]. Consequently, ξf plays a more effective role in improving the spectral purity of the generated microwaves at the regime of ultrashort optical feedback. Hence, even though τf is much shorter for ultrashort optical feedback than long optical feedback, the capability of increasing ξf 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 fl = 1/τf [19, 20, 22, 24]. For long optical feedback where τf 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 τf 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 µm laser section followed by a 100 µm gain/absorption section, a 150 µm 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 τf 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 2π for every 6.2 mA using a 150 µm phase section. The length of either the gain/absorption section or the phase section can be further decreased or increased to acquire a different τf for practical applications of this study. The bias current of either section needs to be adjusted accordingly to achieve a similar level of ξf 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 τf, 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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References

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  1. J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27, 314–335 (2009).
    [Crossref]
  2. X. Q. Qi and J. M. Liu, “Photonic microwave applications of the dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1198–1211 (2011).
    [Crossref]
  3. U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
    [Crossref]
  4. G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7, 118–122 (2013).
    [Crossref]
  5. O. Solgaard and K. Y. Lau, “Optical feedback stabilization of the intensity oscillations in ultrahigh-frequency passively modelocked monolithic quantum-well lasers,” IEEE Photonics Technol. Lett. 5, 1264–1266 (1993).
    [Crossref]
  6. C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
    [Crossref]
  7. X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
    [Crossref]
  8. S. Pan and J. Yao, “Wideband and frequency-tunable microwave generation using an optoelectronic oscillator incorporating a Fabry-Perot laser diode with external optical injection,” Opt. Lett. 35, 1911–1913 (2010).
    [Crossref] [PubMed]
  9. S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-µ m semiconductor lasers subject to strong injection locking,” IEEE Photonics Technol. Lett. 16, 972–974 (2004).
    [Crossref]
  10. E. K. Lau, X. Zhao, H. K. Sung, D. Parekh, C. J. Chang-Hasnain, and M. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100-GHz resonance frequencies and 80-GHz intrinsic bandwidths,” Opt. Express 16, 6609–6618 (2008).
    [Crossref] [PubMed]
  11. H. Chi and J. Yao, “Frequency quadrupling and upconversion in a radio over fiber link,” IEEE J. Lightwave Technol. 26, 2706–2711 (2008).
    [Crossref]
  12. C. T. Lin, P. T. Shih, W. J. Jiang, J. Chen, P. C. Peng, and S. Chi, “A continuously tunable and filterless optical millimeter-wave generation via frequency octupling,” Opt. Express 17, 19749–19756 (2009).
    [Crossref] [PubMed]
  13. T. B. Simpson and F. Doft, “Double-locked laser diode for microwave photonics applications,” IEEE Photonics Technol. Lett. 11, 1476–1478 (1999).
    [Crossref]
  14. S. C. Chan and J. M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10, 1025–1032 (2004).
    [Crossref]
  15. S. C. Chan, S. K. Hwang, and J. M. Liu, “Period-one oscillation for photonic microwave transmission using an optically injected semiconductor laser,” Opt. Express 15, 14921–14935 (2007).
    [Crossref] [PubMed]
  16. M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
    [Crossref]
  17. Y. S. Yuan and F. Y. Lin, “Photonic generation of broadly tunable microwave signals utilizing a dual-beam optically injected semiconductor laser,” IEEE Photonics J. 3, 644–650 (2011).
    [Crossref]
  18. A. Quirce and A. Valle, “High-frequency microwave signal generation using multi-transverse mode VCSELs subject to two-frequency optical injection,” Opt. Express 20, 13390–13401 (2012).
    [Crossref] [PubMed]
  19. J. P. Zhuang and S. C. Chan, “Tunable photonic microwave generation using optically injected semiconductor laser dynamics with optical feedback stabilization,” Opt. Lett. 38, 344–346 (2013).
    [Crossref] [PubMed]
  20. T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Linewidth sharpening via polarization-rotated feedback in optically-injected semiconductor laser oscillators,” IEEE J. Sel. Top. Quantum Electron. 19, 1500807 (2013).
    [Crossref]
  21. A. Hurtado, J. Mee, M. Nami, I. D. Henning, M. J. Adams, and L. F. Lester, “Tunable microwave signal generator with an optically-injected 1310nm QD-DFB laser,” Opt. Express 21, 10772–10778 (2013).
    [Crossref] [PubMed]
  22. K. H. Lo, S. K. Hwang, and S. Donati, “Optical feedback stabilization of photonic microwave generation using period-one nonlinear dynamics of semiconductor lasers,” Opt. Express 22, 18648–18661 (2014).
    [Crossref] [PubMed]
  23. Y. H. Hung and S. K. Hwang, “Photonic microwave stabilization for period-one nonlinear dynamics of semiconductor lasers using optical modulation sideband injection locking,” Opt. Express,  23, 6520–6532 (2015).
    [Crossref] [PubMed]
  24. J. P. Zhuang and S. C. Chan, “Phase noise characteristics of microwave signals generated by semiconductor laser dynamics,” Opt. Express 23, 2777–2797 (2015).
    [Crossref] [PubMed]
  25. C. Wang, R. Raghunathan, K. Schires, S. C. Chan, L. F. Lester, and F. Grillot, “Optically injected InAs/GaAs quantum dot laser for tunable photonic microwave generation,” Opt. Lett. 41, 1153–1156 (2016).
    [Crossref] [PubMed]
  26. L. Fan, G. Xia, J. Chen, X. Tang, Q. Liang, and Z. Wu, “High-purity 60GHz band millimeter-wave generation based on optically injected semiconductor laser under subharmonic microwave modulation,” Opt. Express,  24, 18252–18265 (2016).
    [Crossref] [PubMed]
  27. J. S. Suelzer, T. B. Simpson, P. Devgan, and N. G. Usechak, “Tunable, low-phase-noise microwave signals from an optically injected semiconductor laser with opto-electronic feedback,” Opt. Lett. 42, 3181–3184 (2017).
    [Crossref] [PubMed]
  28. T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt. 9, 765–784 (1997).
    [Crossref]
  29. S. K. Hwang, J. M. Liu, and J. K. White, “Characteristics of period-one oscillations in semiconductor lasers subject to optical injection,” IEEE J. Sel. Top. Quantum Electron. 10, 974–981 (2004).
    [Crossref]
  30. S. Donati and S. K. Hwang, “Chaos and high-level dynamics in coupled lasers and their applications,” Prog. Quantum Electron. 36, 293–341 (2012).
    [Crossref]
  31. S. K. Hwang, S. C. Chan, S. C. Hsieh, and C.Y. Li, “Photonic microwave generation and transmission using direct modulation of stably injection-locked semiconductor lasers,” Opt. Commun. 284, 3581–3589 (2011).
    [Crossref]
  32. T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A 53, 4372–4380 (1996).
    [Crossref] [PubMed]
  33. B. Krauskopf, N. Tollenaar, and D. Lenstra, “Tori and their bifurcations in an optically injected semiconductor laser,” Opt. Commun. 156, 158–169 (1998).
    [Crossref]
  34. S. K. Hwang and D. H. Liang, “Effects of linewidth enhancement factor on period-one oscillations of optically injected semiconductor lasers,” Appl. Phys. Lett. 89, 061120 (2006).
    [Crossref]
  35. S. C. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46, 421–428 (2010).
    [Crossref]
  36. T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Limit-cycle dynamics with reduced sensitivity to perturbations,” Phys. Rev. Lett. 112, 023901 (2014).
    [Crossref] [PubMed]
  37. C. J. Lin, M. AlMulla, and J. M. Liu, “Harmonic analysis of limit-cycle oscillations of an optically injected semiconductor laser,” IEEE J. Quantum Electron. 50, 815–822 (2014).
  38. S. C. Chan, S. K. Hwang, and J. M. Liu, “Radio-over-fiber AM-to-FM upconversion using an optically injected semiconductor laser,” Opt. Lett. 31, 2254–2256 (2006).
    [Crossref] [PubMed]
  39. R. Diaz, S. C. Chan, and J. M. Liu, “Lidar detection using a dual-frequency source,” Opt. Lett. 31, 3600–3602 (2006).
    [Crossref] [PubMed]
  40. S. K. Hwang, H. F. Chen, and C. Y. Lin, “All-optical frequency conversion using nonlinear dynamics of semiconductor lasers,” Opt. Lett. 34, 812–814 (2009).
    [Crossref] [PubMed]
  41. C. Cui, X. Fu, and S. C. Chan, “Double-locked semiconductor laser for radio-over-fiber uplink transmission,” Opt. Lett. 34, 3821–3823 (2009).
    [Crossref] [PubMed]
  42. C. H. Chu, S. L. Lin, S. C. Chan, and S. K. Hwang, “All-optical modulation format conversion using nonlinear dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 48, 1389–1396 (2012).
    [Crossref]
  43. Y. H. Hung, C. H. Chu, and S. K. Hwang, “Optical double-sideband modulation to single-sideband modulation conversion using period-one nonlinear dynamics of semiconductor lasers for radio-over-fiber links,” Opt. Lett. 38, 1482–1484 (2013).
    [Crossref] [PubMed]
  44. Y. H. Hung and S. K. Hwang, “Photonic microwave amplification for radio-over-fiber links using period-one nonlinear dynamics of semiconductor lasers,” Opt. Lett. 38, 3355–3358 (2013).
    [Crossref] [PubMed]
  45. P. Zhou, F. Z. Zhang, Q. S. Guo, and S. L. Pan, “Linearly chirped microwave waveform generation with large time-bandwidth product by optically injected semiconductor laser,” Opt. Express 24, 18460–18467 (2016).
    [Crossref] [PubMed]
  46. J. P. Zhuang, X. Z. Li, S. S. Li, and S. C. Chan, “Frequency-modulated microwave generation with feedback stabilization using an optically injected semiconductor laser,” Opt. Lett. 41, 5764–5767 (2016).
    [Crossref] [PubMed]
  47. Y. H. Hung, J. H. Yan, K. M. Feng, and S. K. Hwang, “Photonic microwave carrier recovery using period-one nonlinear dynamics of semiconductor lasers for OFDM-RoF coherent detection,” Opt. Lett. 42, 2402–2405 (2017).
    [Crossref] [PubMed]
  48. K. L. Hsieh, S. K. Hwang, and C. L. Yang, “Photonic microwave time delay using slow- and fast-light effects in optically injected semiconductor lasers,” Opt. Lett. 42, 3307–3310 (2017).
    [Crossref] [PubMed]
  49. O. Ushakov, S. Bauer, O. Brox, H.-J. Wunsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
    [Crossref] [PubMed]
  50. A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
    [Crossref] [PubMed]
  51. A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).
  52. A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18, 5188–5198 (2010).
    [Crossref] [PubMed]
  53. R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22, 11727–11740 (2014).
    [Crossref] [PubMed]

2017 (4)

2016 (4)

2015 (2)

2014 (4)

K. H. Lo, S. K. Hwang, and S. Donati, “Optical feedback stabilization of photonic microwave generation using period-one nonlinear dynamics of semiconductor lasers,” Opt. Express 22, 18648–18661 (2014).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Limit-cycle dynamics with reduced sensitivity to perturbations,” Phys. Rev. Lett. 112, 023901 (2014).
[Crossref] [PubMed]

C. J. Lin, M. AlMulla, and J. M. Liu, “Harmonic analysis of limit-cycle oscillations of an optically injected semiconductor laser,” IEEE J. Quantum Electron. 50, 815–822 (2014).

R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22, 11727–11740 (2014).
[Crossref] [PubMed]

2013 (6)

2012 (3)

A. Quirce and A. Valle, “High-frequency microwave signal generation using multi-transverse mode VCSELs subject to two-frequency optical injection,” Opt. Express 20, 13390–13401 (2012).
[Crossref] [PubMed]

S. Donati and S. K. Hwang, “Chaos and high-level dynamics in coupled lasers and their applications,” Prog. Quantum Electron. 36, 293–341 (2012).
[Crossref]

C. H. Chu, S. L. Lin, S. C. Chan, and S. K. Hwang, “All-optical modulation format conversion using nonlinear dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 48, 1389–1396 (2012).
[Crossref]

2011 (3)

S. K. Hwang, S. C. Chan, S. C. Hsieh, and C.Y. Li, “Photonic microwave generation and transmission using direct modulation of stably injection-locked semiconductor lasers,” Opt. Commun. 284, 3581–3589 (2011).
[Crossref]

Y. S. Yuan and F. Y. Lin, “Photonic generation of broadly tunable microwave signals utilizing a dual-beam optically injected semiconductor laser,” IEEE Photonics J. 3, 644–650 (2011).
[Crossref]

X. Q. Qi and J. M. Liu, “Photonic microwave applications of the dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1198–1211 (2011).
[Crossref]

2010 (5)

C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
[Crossref]

S. Pan and J. Yao, “Wideband and frequency-tunable microwave generation using an optoelectronic oscillator incorporating a Fabry-Perot laser diode with external optical injection,” Opt. Lett. 35, 1911–1913 (2010).
[Crossref] [PubMed]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
[Crossref]

S. C. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46, 421–428 (2010).
[Crossref]

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18, 5188–5198 (2010).
[Crossref] [PubMed]

2009 (4)

2008 (3)

E. K. Lau, X. Zhao, H. K. Sung, D. Parekh, C. J. Chang-Hasnain, and M. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100-GHz resonance frequencies and 80-GHz intrinsic bandwidths,” Opt. Express 16, 6609–6618 (2008).
[Crossref] [PubMed]

H. Chi and J. Yao, “Frequency quadrupling and upconversion in a radio over fiber link,” IEEE J. Lightwave Technol. 26, 2706–2711 (2008).
[Crossref]

A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (3)

2004 (4)

O. Ushakov, S. Bauer, O. Brox, H.-J. Wunsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

S. K. Hwang, J. M. Liu, and J. K. White, “Characteristics of period-one oscillations in semiconductor lasers subject to optical injection,” IEEE J. Sel. Top. Quantum Electron. 10, 974–981 (2004).
[Crossref]

S. C. Chan and J. M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10, 1025–1032 (2004).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-µ m semiconductor lasers subject to strong injection locking,” IEEE Photonics Technol. Lett. 16, 972–974 (2004).
[Crossref]

1999 (1)

T. B. Simpson and F. Doft, “Double-locked laser diode for microwave photonics applications,” IEEE Photonics Technol. Lett. 11, 1476–1478 (1999).
[Crossref]

1998 (1)

B. Krauskopf, N. Tollenaar, and D. Lenstra, “Tori and their bifurcations in an optically injected semiconductor laser,” Opt. Commun. 156, 158–169 (1998).
[Crossref]

1997 (1)

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt. 9, 765–784 (1997).
[Crossref]

1996 (2)

T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A 53, 4372–4380 (1996).
[Crossref] [PubMed]

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

1993 (1)

O. Solgaard and K. Y. Lau, “Optical feedback stabilization of the intensity oscillations in ultrahigh-frequency passively modelocked monolithic quantum-well lasers,” IEEE Photonics Technol. Lett. 5, 1264–1266 (1993).
[Crossref]

1992 (1)

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Adams, M. J.

Akizawa, Y.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22, 11727–11740 (2014).
[Crossref] [PubMed]

AlMulla, M.

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Limit-cycle dynamics with reduced sensitivity to perturbations,” Phys. Rev. Lett. 112, 023901 (2014).
[Crossref] [PubMed]

C. J. Lin, M. AlMulla, and J. M. Liu, “Harmonic analysis of limit-cycle oscillations of an optically injected semiconductor laser,” IEEE J. Quantum Electron. 50, 815–822 (2014).

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Linewidth sharpening via polarization-rotated feedback in optically-injected semiconductor laser oscillators,” IEEE J. Sel. Top. Quantum Electron. 19, 1500807 (2013).
[Crossref]

Alsing, P. M.

T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A 53, 4372–4380 (1996).
[Crossref] [PubMed]

Arai, K.

Argyris, A.

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18, 5188–5198 (2010).
[Crossref] [PubMed]

A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Bauer, S.

O. Ushakov, S. Bauer, O. Brox, H.-J. Wunsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Bogris, A.

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18, 5188–5198 (2010).
[Crossref] [PubMed]

A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Bosco, D.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

Broberg, B.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Brox, O.

O. Ushakov, S. Bauer, O. Brox, H.-J. Wunsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Bruun, M.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Chan, S. C.

C. Wang, R. Raghunathan, K. Schires, S. C. Chan, L. F. Lester, and F. Grillot, “Optically injected InAs/GaAs quantum dot laser for tunable photonic microwave generation,” Opt. Lett. 41, 1153–1156 (2016).
[Crossref] [PubMed]

J. P. Zhuang, X. Z. Li, S. S. Li, and S. C. Chan, “Frequency-modulated microwave generation with feedback stabilization using an optically injected semiconductor laser,” Opt. Lett. 41, 5764–5767 (2016).
[Crossref] [PubMed]

J. P. Zhuang and S. C. Chan, “Phase noise characteristics of microwave signals generated by semiconductor laser dynamics,” Opt. Express 23, 2777–2797 (2015).
[Crossref] [PubMed]

J. P. Zhuang and S. C. Chan, “Tunable photonic microwave generation using optically injected semiconductor laser dynamics with optical feedback stabilization,” Opt. Lett. 38, 344–346 (2013).
[Crossref] [PubMed]

C. H. Chu, S. L. Lin, S. C. Chan, and S. K. Hwang, “All-optical modulation format conversion using nonlinear dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 48, 1389–1396 (2012).
[Crossref]

S. K. Hwang, S. C. Chan, S. C. Hsieh, and C.Y. Li, “Photonic microwave generation and transmission using direct modulation of stably injection-locked semiconductor lasers,” Opt. Commun. 284, 3581–3589 (2011).
[Crossref]

S. C. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46, 421–428 (2010).
[Crossref]

C. Cui, X. Fu, and S. C. Chan, “Double-locked semiconductor laser for radio-over-fiber uplink transmission,” Opt. Lett. 34, 3821–3823 (2009).
[Crossref] [PubMed]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Period-one oscillation for photonic microwave transmission using an optically injected semiconductor laser,” Opt. Express 15, 14921–14935 (2007).
[Crossref] [PubMed]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Radio-over-fiber AM-to-FM upconversion using an optically injected semiconductor laser,” Opt. Lett. 31, 2254–2256 (2006).
[Crossref] [PubMed]

R. Diaz, S. C. Chan, and J. M. Liu, “Lidar detection using a dual-frequency source,” Opt. Lett. 31, 3600–3602 (2006).
[Crossref] [PubMed]

S. C. Chan and J. M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10, 1025–1032 (2004).
[Crossref]

Chang-Hasnain, C. J.

Chen, H. F.

Chen, J.

Chi, H.

H. Chi and J. Yao, “Frequency quadrupling and upconversion in a radio over fiber link,” IEEE J. Lightwave Technol. 26, 2706–2711 (2008).
[Crossref]

Chi, S.

Chlouverakis, K. E.

A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Christensen, E. L.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Chu, C. H.

Y. H. Hung, C. H. Chu, and S. K. Hwang, “Optical double-sideband modulation to single-sideband modulation conversion using period-one nonlinear dynamics of semiconductor lasers for radio-over-fiber links,” Opt. Lett. 38, 1482–1484 (2013).
[Crossref] [PubMed]

C. H. Chu, S. L. Lin, S. C. Chan, and S. K. Hwang, “All-optical modulation format conversion using nonlinear dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 48, 1389–1396 (2012).
[Crossref]

Cui, C.

Davis, P.

Devgan, P.

Diaz, R.

Doft, F.

T. B. Simpson and F. Doft, “Double-locked laser diode for microwave photonics applications,” IEEE Photonics Technol. Lett. 11, 1476–1478 (1999).
[Crossref]

Donati, S.

Erneux, T.

T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A 53, 4372–4380 (1996).
[Crossref] [PubMed]

Fan, L.

Feng, K. M.

Fu, X.

Gavrielides, A.

T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A 53, 4372–4380 (1996).
[Crossref] [PubMed]

Gliese, U.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Grillot, F.

C. Wang, R. Raghunathan, K. Schires, S. C. Chan, L. F. Lester, and F. Grillot, “Optically injected InAs/GaAs quantum dot laser for tunable photonic microwave generation,” Opt. Lett. 41, 1153–1156 (2016).
[Crossref] [PubMed]

C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
[Crossref]

Grivas, E.

Guo, Q. S.

Hamacher, M.

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18, 5188–5198 (2010).
[Crossref] [PubMed]

A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Harayama, T.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22, 11727–11740 (2014).
[Crossref] [PubMed]

Henneberger, F.

O. Ushakov, S. Bauer, O. Brox, H.-J. Wunsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Henning, I. D.

Hsieh, K. L.

Hsieh, S. C.

S. K. Hwang, S. C. Chan, S. C. Hsieh, and C.Y. Li, “Photonic microwave generation and transmission using direct modulation of stably injection-locked semiconductor lasers,” Opt. Commun. 284, 3581–3589 (2011).
[Crossref]

Huang, K. F.

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt. 9, 765–784 (1997).
[Crossref]

Hung, Y. H.

Hurtado, A.

Hwang, S. K.

K. L. Hsieh, S. K. Hwang, and C. L. Yang, “Photonic microwave time delay using slow- and fast-light effects in optically injected semiconductor lasers,” Opt. Lett. 42, 3307–3310 (2017).
[Crossref] [PubMed]

Y. H. Hung, J. H. Yan, K. M. Feng, and S. K. Hwang, “Photonic microwave carrier recovery using period-one nonlinear dynamics of semiconductor lasers for OFDM-RoF coherent detection,” Opt. Lett. 42, 2402–2405 (2017).
[Crossref] [PubMed]

Y. H. Hung and S. K. Hwang, “Photonic microwave stabilization for period-one nonlinear dynamics of semiconductor lasers using optical modulation sideband injection locking,” Opt. Express,  23, 6520–6532 (2015).
[Crossref] [PubMed]

K. H. Lo, S. K. Hwang, and S. Donati, “Optical feedback stabilization of photonic microwave generation using period-one nonlinear dynamics of semiconductor lasers,” Opt. Express 22, 18648–18661 (2014).
[Crossref] [PubMed]

Y. H. Hung, C. H. Chu, and S. K. Hwang, “Optical double-sideband modulation to single-sideband modulation conversion using period-one nonlinear dynamics of semiconductor lasers for radio-over-fiber links,” Opt. Lett. 38, 1482–1484 (2013).
[Crossref] [PubMed]

Y. H. Hung and S. K. Hwang, “Photonic microwave amplification for radio-over-fiber links using period-one nonlinear dynamics of semiconductor lasers,” Opt. Lett. 38, 3355–3358 (2013).
[Crossref] [PubMed]

S. Donati and S. K. Hwang, “Chaos and high-level dynamics in coupled lasers and their applications,” Prog. Quantum Electron. 36, 293–341 (2012).
[Crossref]

C. H. Chu, S. L. Lin, S. C. Chan, and S. K. Hwang, “All-optical modulation format conversion using nonlinear dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 48, 1389–1396 (2012).
[Crossref]

S. K. Hwang, S. C. Chan, S. C. Hsieh, and C.Y. Li, “Photonic microwave generation and transmission using direct modulation of stably injection-locked semiconductor lasers,” Opt. Commun. 284, 3581–3589 (2011).
[Crossref]

S. K. Hwang, H. F. Chen, and C. Y. Lin, “All-optical frequency conversion using nonlinear dynamics of semiconductor lasers,” Opt. Lett. 34, 812–814 (2009).
[Crossref] [PubMed]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Period-one oscillation for photonic microwave transmission using an optically injected semiconductor laser,” Opt. Express 15, 14921–14935 (2007).
[Crossref] [PubMed]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Radio-over-fiber AM-to-FM upconversion using an optically injected semiconductor laser,” Opt. Lett. 31, 2254–2256 (2006).
[Crossref] [PubMed]

S. K. Hwang and D. H. Liang, “Effects of linewidth enhancement factor on period-one oscillations of optically injected semiconductor lasers,” Appl. Phys. Lett. 89, 061120 (2006).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “Characteristics of period-one oscillations in semiconductor lasers subject to optical injection,” IEEE J. Sel. Top. Quantum Electron. 10, 974–981 (2004).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-µ m semiconductor lasers subject to strong injection locking,” IEEE Photonics Technol. Lett. 16, 972–974 (2004).
[Crossref]

Inubushi, M.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

Jiang, W. J.

Karsaklian, A.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

Kovanis, V.

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Limit-cycle dynamics with reduced sensitivity to perturbations,” Phys. Rev. Lett. 112, 023901 (2014).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Linewidth sharpening via polarization-rotated feedback in optically-injected semiconductor laser oscillators,” IEEE J. Sel. Top. Quantum Electron. 19, 1500807 (2013).
[Crossref]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
[Crossref]

T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A 53, 4372–4380 (1996).
[Crossref] [PubMed]

Krauskopf, B.

B. Krauskopf, N. Tollenaar, and D. Lenstra, “Tori and their bifurcations in an optically injected semiconductor laser,” Opt. Commun. 156, 158–169 (1998).
[Crossref]

Lau, E. K.

Lau, K. Y.

O. Solgaard and K. Y. Lau, “Optical feedback stabilization of the intensity oscillations in ultrahigh-frequency passively modelocked monolithic quantum-well lasers,” IEEE Photonics Technol. Lett. 5, 1264–1266 (1993).
[Crossref]

Lenstra, D.

B. Krauskopf, N. Tollenaar, and D. Lenstra, “Tori and their bifurcations in an optically injected semiconductor laser,” Opt. Commun. 156, 158–169 (1998).
[Crossref]

Lester, L. F.

C. Wang, R. Raghunathan, K. Schires, S. C. Chan, L. F. Lester, and F. Grillot, “Optically injected InAs/GaAs quantum dot laser for tunable photonic microwave generation,” Opt. Lett. 41, 1153–1156 (2016).
[Crossref] [PubMed]

A. Hurtado, J. Mee, M. Nami, I. D. Henning, M. J. Adams, and L. F. Lester, “Tunable microwave signal generator with an optically-injected 1310nm QD-DFB laser,” Opt. Express 21, 10772–10778 (2013).
[Crossref] [PubMed]

C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
[Crossref]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
[Crossref]

Li, C.Y.

S. K. Hwang, S. C. Chan, S. C. Hsieh, and C.Y. Li, “Photonic microwave generation and transmission using direct modulation of stably injection-locked semiconductor lasers,” Opt. Commun. 284, 3581–3589 (2011).
[Crossref]

Li, S. S.

Li, X. Z.

Li, Y.

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
[Crossref]

C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
[Crossref]

Liang, D. H.

S. K. Hwang and D. H. Liang, “Effects of linewidth enhancement factor on period-one oscillations of optically injected semiconductor lasers,” Appl. Phys. Lett. 89, 061120 (2006).
[Crossref]

Liang, Q.

Lin, C. J.

C. J. Lin, M. AlMulla, and J. M. Liu, “Harmonic analysis of limit-cycle oscillations of an optically injected semiconductor laser,” IEEE J. Quantum Electron. 50, 815–822 (2014).

Lin, C. T.

Lin, C. Y.

C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
[Crossref]

S. K. Hwang, H. F. Chen, and C. Y. Lin, “All-optical frequency conversion using nonlinear dynamics of semiconductor lasers,” Opt. Lett. 34, 812–814 (2009).
[Crossref] [PubMed]

Lin, F. Y.

Y. S. Yuan and F. Y. Lin, “Photonic generation of broadly tunable microwave signals utilizing a dual-beam optically injected semiconductor laser,” IEEE Photonics J. 3, 644–650 (2011).
[Crossref]

Lin, S. L.

C. H. Chu, S. L. Lin, S. C. Chan, and S. K. Hwang, “All-optical modulation format conversion using nonlinear dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 48, 1389–1396 (2012).
[Crossref]

Lindgren, S.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Liu, J. M.

C. J. Lin, M. AlMulla, and J. M. Liu, “Harmonic analysis of limit-cycle oscillations of an optically injected semiconductor laser,” IEEE J. Quantum Electron. 50, 815–822 (2014).

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Limit-cycle dynamics with reduced sensitivity to perturbations,” Phys. Rev. Lett. 112, 023901 (2014).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Linewidth sharpening via polarization-rotated feedback in optically-injected semiconductor laser oscillators,” IEEE J. Sel. Top. Quantum Electron. 19, 1500807 (2013).
[Crossref]

X. Q. Qi and J. M. Liu, “Photonic microwave applications of the dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1198–1211 (2011).
[Crossref]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Period-one oscillation for photonic microwave transmission using an optically injected semiconductor laser,” Opt. Express 15, 14921–14935 (2007).
[Crossref] [PubMed]

R. Diaz, S. C. Chan, and J. M. Liu, “Lidar detection using a dual-frequency source,” Opt. Lett. 31, 3600–3602 (2006).
[Crossref] [PubMed]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Radio-over-fiber AM-to-FM upconversion using an optically injected semiconductor laser,” Opt. Lett. 31, 2254–2256 (2006).
[Crossref] [PubMed]

S. K. Hwang, J. M. Liu, and J. K. White, “Characteristics of period-one oscillations in semiconductor lasers subject to optical injection,” IEEE J. Sel. Top. Quantum Electron. 10, 974–981 (2004).
[Crossref]

S. C. Chan and J. M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10, 1025–1032 (2004).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-µ m semiconductor lasers subject to strong injection locking,” IEEE Photonics Technol. Lett. 16, 972–974 (2004).
[Crossref]

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt. 9, 765–784 (1997).
[Crossref]

Lo, K. H.

Maleki, L.

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

Mee, J.

Murakowski, J. A.

G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7, 118–122 (2013).
[Crossref]

Naderi, N. A.

C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
[Crossref]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
[Crossref]

Nami, M.

Nielsen, T. N.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Ohara, S.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

Pan, S.

Pan, S. L.

Parekh, D.

Peng, P. C.

Pochet, M.

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
[Crossref]

Prather, D. W.

G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7, 118–122 (2013).
[Crossref]

Qi, X. Q.

X. Q. Qi and J. M. Liu, “Photonic microwave applications of the dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1198–1211 (2011).
[Crossref]

Quirce, A.

Raghunathan, R.

Sato, N.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

Schires, K.

Schneider, G. J.

G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7, 118–122 (2013).
[Crossref]

Schuetz, C. A.

G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7, 118–122 (2013).
[Crossref]

Shi, S.

G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7, 118–122 (2013).
[Crossref]

Shih, P. T.

Simpson, T. B.

J. S. Suelzer, T. B. Simpson, P. Devgan, and N. G. Usechak, “Tunable, low-phase-noise microwave signals from an optically injected semiconductor laser with opto-electronic feedback,” Opt. Lett. 42, 3181–3184 (2017).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Limit-cycle dynamics with reduced sensitivity to perturbations,” Phys. Rev. Lett. 112, 023901 (2014).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Linewidth sharpening via polarization-rotated feedback in optically-injected semiconductor laser oscillators,” IEEE J. Sel. Top. Quantum Electron. 19, 1500807 (2013).
[Crossref]

T. B. Simpson and F. Doft, “Double-locked laser diode for microwave photonics applications,” IEEE Photonics Technol. Lett. 11, 1476–1478 (1999).
[Crossref]

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt. 9, 765–784 (1997).
[Crossref]

Solgaard, O.

O. Solgaard and K. Y. Lau, “Optical feedback stabilization of the intensity oscillations in ultrahigh-frequency passively modelocked monolithic quantum-well lasers,” IEEE Photonics Technol. Lett. 5, 1264–1266 (1993).
[Crossref]

Stubkjaer, K. E.

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

Suelzer, J. S.

Sunada, S.

Sung, H. K.

Syvridis, D.

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18, 5188–5198 (2010).
[Crossref] [PubMed]

A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Tai, K.

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt. 9, 765–784 (1997).
[Crossref]

Takahashi, R.

Tang, X.

Tollenaar, N.

B. Krauskopf, N. Tollenaar, and D. Lenstra, “Tori and their bifurcations in an optically injected semiconductor laser,” Opt. Commun. 156, 158–169 (1998).
[Crossref]

Uchida, A.

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22, 11727–11740 (2014).
[Crossref] [PubMed]

Usechak, N. G.

J. S. Suelzer, T. B. Simpson, P. Devgan, and N. G. Usechak, “Tunable, low-phase-noise microwave signals from an optically injected semiconductor laser with opto-electronic feedback,” Opt. Lett. 42, 3181–3184 (2017).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Limit-cycle dynamics with reduced sensitivity to perturbations,” Phys. Rev. Lett. 112, 023901 (2014).
[Crossref] [PubMed]

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Linewidth sharpening via polarization-rotated feedback in optically-injected semiconductor laser oscillators,” IEEE J. Sel. Top. Quantum Electron. 19, 1500807 (2013).
[Crossref]

Ushakov, O.

O. Ushakov, S. Bauer, O. Brox, H.-J. Wunsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Valle, A.

Wang, C.

White, J. K.

S. K. Hwang, J. M. Liu, and J. K. White, “Characteristics of period-one oscillations in semiconductor lasers subject to optical injection,” IEEE J. Sel. Top. Quantum Electron. 10, 974–981 (2004).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-µ m semiconductor lasers subject to strong injection locking,” IEEE Photonics Technol. Lett. 16, 972–974 (2004).
[Crossref]

Wu, M.

Wu, Z.

Wunsche, H.-J.

O. Ushakov, S. Bauer, O. Brox, H.-J. Wunsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Xia, G.

Yan, J. H.

Yang, C. L.

Yao, J.

Yao, J. P.

Yao, X. S.

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

Yoshimura, K.

Yuan, Y. S.

Y. S. Yuan and F. Y. Lin, “Photonic generation of broadly tunable microwave signals utilizing a dual-beam optically injected semiconductor laser,” IEEE Photonics J. 3, 644–650 (2011).
[Crossref]

Zhang, F. Z.

Zhao, X.

Zhou, P.

Zhuang, J. P.

Appl. Phys. Lett. (2)

C. Y. Lin, F. Grillot, N. A. Naderi, Y. Li, and L. F. Lester, “rf linewidth reduction in a quantum dot passively mode-locked laser subject to external optical feedback,” Appl. Phys. Lett. 96, 051118 (2010).
[Crossref]

S. K. Hwang and D. H. Liang, “Effects of linewidth enhancement factor on period-one oscillations of optically injected semiconductor lasers,” Appl. Phys. Lett. 89, 061120 (2006).
[Crossref]

IEEE J. Lightwave Technol. (1)

H. Chi and J. Yao, “Frequency quadrupling and upconversion in a radio over fiber link,” IEEE J. Lightwave Technol. 26, 2706–2711 (2008).
[Crossref]

IEEE J. Quantum Electron. (4)

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

S. C. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46, 421–428 (2010).
[Crossref]

C. J. Lin, M. AlMulla, and J. M. Liu, “Harmonic analysis of limit-cycle oscillations of an optically injected semiconductor laser,” IEEE J. Quantum Electron. 50, 815–822 (2014).

C. H. Chu, S. L. Lin, S. C. Chan, and S. K. Hwang, “All-optical modulation format conversion using nonlinear dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 48, 1389–1396 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (4)

T. B. Simpson, J. M. Liu, M. AlMulla, N. G. Usechak, and V. Kovanis, “Linewidth sharpening via polarization-rotated feedback in optically-injected semiconductor laser oscillators,” IEEE J. Sel. Top. Quantum Electron. 19, 1500807 (2013).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “Characteristics of period-one oscillations in semiconductor lasers subject to optical injection,” IEEE J. Sel. Top. Quantum Electron. 10, 974–981 (2004).
[Crossref]

X. Q. Qi and J. M. Liu, “Photonic microwave applications of the dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1198–1211 (2011).
[Crossref]

S. C. Chan and J. M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10, 1025–1032 (2004).
[Crossref]

IEEE Photonics J. (2)

Y. S. Yuan and F. Y. Lin, “Photonic generation of broadly tunable microwave signals utilizing a dual-beam optically injected semiconductor laser,” IEEE Photonics J. 3, 644–650 (2011).
[Crossref]

A. Karsaklian, D. Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9, 6600512 (2017).

IEEE Photonics Technol. Lett. (5)

T. B. Simpson and F. Doft, “Double-locked laser diode for microwave photonics applications,” IEEE Photonics Technol. Lett. 11, 1476–1478 (1999).
[Crossref]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, and L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22, 763–765 (2010).
[Crossref]

U. Gliese, T. N. Nielsen, M. Bruun, E. L. Christensen, K. E. Stubkjaer, S. Lindgren, and B. Broberg, “A wideband heterodyne optical phase-locked loop for generation of 3–18 GHz Microwave Carriers,” IEEE Photonics Technol. Lett. 4, 936–938 (1992).
[Crossref]

O. Solgaard and K. Y. Lau, “Optical feedback stabilization of the intensity oscillations in ultrahigh-frequency passively modelocked monolithic quantum-well lasers,” IEEE Photonics Technol. Lett. 5, 1264–1266 (1993).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-µ m semiconductor lasers subject to strong injection locking,” IEEE Photonics Technol. Lett. 16, 972–974 (2004).
[Crossref]

J. Lightwave Technol. (1)

Nat. Photonics (1)

G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7, 118–122 (2013).
[Crossref]

Opt. Commun. (2)

S. K. Hwang, S. C. Chan, S. C. Hsieh, and C.Y. Li, “Photonic microwave generation and transmission using direct modulation of stably injection-locked semiconductor lasers,” Opt. Commun. 284, 3581–3589 (2011).
[Crossref]

B. Krauskopf, N. Tollenaar, and D. Lenstra, “Tori and their bifurcations in an optically injected semiconductor laser,” Opt. Commun. 156, 158–169 (1998).
[Crossref]

Opt. Express (12)

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18, 5188–5198 (2010).
[Crossref] [PubMed]

R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22, 11727–11740 (2014).
[Crossref] [PubMed]

P. Zhou, F. Z. Zhang, Q. S. Guo, and S. L. Pan, “Linearly chirped microwave waveform generation with large time-bandwidth product by optically injected semiconductor laser,” Opt. Express 24, 18460–18467 (2016).
[Crossref] [PubMed]

L. Fan, G. Xia, J. Chen, X. Tang, Q. Liang, and Z. Wu, “High-purity 60GHz band millimeter-wave generation based on optically injected semiconductor laser under subharmonic microwave modulation,” Opt. Express,  24, 18252–18265 (2016).
[Crossref] [PubMed]

A. Hurtado, J. Mee, M. Nami, I. D. Henning, M. J. Adams, and L. F. Lester, “Tunable microwave signal generator with an optically-injected 1310nm QD-DFB laser,” Opt. Express 21, 10772–10778 (2013).
[Crossref] [PubMed]

K. H. Lo, S. K. Hwang, and S. Donati, “Optical feedback stabilization of photonic microwave generation using period-one nonlinear dynamics of semiconductor lasers,” Opt. Express 22, 18648–18661 (2014).
[Crossref] [PubMed]

Y. H. Hung and S. K. Hwang, “Photonic microwave stabilization for period-one nonlinear dynamics of semiconductor lasers using optical modulation sideband injection locking,” Opt. Express,  23, 6520–6532 (2015).
[Crossref] [PubMed]

J. P. Zhuang and S. C. Chan, “Phase noise characteristics of microwave signals generated by semiconductor laser dynamics,” Opt. Express 23, 2777–2797 (2015).
[Crossref] [PubMed]

E. K. Lau, X. Zhao, H. K. Sung, D. Parekh, C. J. Chang-Hasnain, and M. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100-GHz resonance frequencies and 80-GHz intrinsic bandwidths,” Opt. Express 16, 6609–6618 (2008).
[Crossref] [PubMed]

A. Quirce and A. Valle, “High-frequency microwave signal generation using multi-transverse mode VCSELs subject to two-frequency optical injection,” Opt. Express 20, 13390–13401 (2012).
[Crossref] [PubMed]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Period-one oscillation for photonic microwave transmission using an optically injected semiconductor laser,” Opt. Express 15, 14921–14935 (2007).
[Crossref] [PubMed]

C. T. Lin, P. T. Shih, W. J. Jiang, J. Chen, P. C. Peng, and S. Chi, “A continuously tunable and filterless optical millimeter-wave generation via frequency octupling,” Opt. Express 17, 19749–19756 (2009).
[Crossref] [PubMed]

Opt. Lett. (13)

J. P. Zhuang and S. C. Chan, “Tunable photonic microwave generation using optically injected semiconductor laser dynamics with optical feedback stabilization,” Opt. Lett. 38, 344–346 (2013).
[Crossref] [PubMed]

S. Pan and J. Yao, “Wideband and frequency-tunable microwave generation using an optoelectronic oscillator incorporating a Fabry-Perot laser diode with external optical injection,” Opt. Lett. 35, 1911–1913 (2010).
[Crossref] [PubMed]

C. Wang, R. Raghunathan, K. Schires, S. C. Chan, L. F. Lester, and F. Grillot, “Optically injected InAs/GaAs quantum dot laser for tunable photonic microwave generation,” Opt. Lett. 41, 1153–1156 (2016).
[Crossref] [PubMed]

J. S. Suelzer, T. B. Simpson, P. Devgan, and N. G. Usechak, “Tunable, low-phase-noise microwave signals from an optically injected semiconductor laser with opto-electronic feedback,” Opt. Lett. 42, 3181–3184 (2017).
[Crossref] [PubMed]

S. C. Chan, S. K. Hwang, and J. M. Liu, “Radio-over-fiber AM-to-FM upconversion using an optically injected semiconductor laser,” Opt. Lett. 31, 2254–2256 (2006).
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R. Diaz, S. C. Chan, and J. M. Liu, “Lidar detection using a dual-frequency source,” Opt. Lett. 31, 3600–3602 (2006).
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S. K. Hwang, H. F. Chen, and C. Y. Lin, “All-optical frequency conversion using nonlinear dynamics of semiconductor lasers,” Opt. Lett. 34, 812–814 (2009).
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J. P. Zhuang, X. Z. Li, S. S. Li, and S. C. Chan, “Frequency-modulated microwave generation with feedback stabilization using an optically injected semiconductor laser,” Opt. Lett. 41, 5764–5767 (2016).
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Y. H. Hung, J. H. Yan, K. M. Feng, and S. K. Hwang, “Photonic microwave carrier recovery using period-one nonlinear dynamics of semiconductor lasers for OFDM-RoF coherent detection,” Opt. Lett. 42, 2402–2405 (2017).
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K. L. Hsieh, S. K. Hwang, and C. L. Yang, “Photonic microwave time delay using slow- and fast-light effects in optically injected semiconductor lasers,” Opt. Lett. 42, 3307–3310 (2017).
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Y. H. Hung and S. K. Hwang, “Photonic microwave amplification for radio-over-fiber links using period-one nonlinear dynamics of semiconductor lasers,” Opt. Lett. 38, 3355–3358 (2013).
[Crossref] [PubMed]

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[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Optical spectra (left column) and microwave spectra (right column) of the laser subject to both optical injection at (ξi, fi) = (0.4, 40 GHz) and optical feedback at (ξf, τf) = (0, 0 ps) for (a–i)(a–ii), (0.1, 0.95 ps) for (b–i)(b–ii), (0.4, 0.95 ps) for (c–i)(c–ii), and (0.1, 23.5 ps) for (d–i)(d–ii), respectively. Red curves, no laser noise is considered; Gray curves, laser noise is considered. The x axes of the optical spectra are relative to the free-running frequency of the laser. Each inset is an enlargement of each corresponding microwave spectrum in a linear scale with a Lorentzian fitting curve in blue.
Fig. 2
Fig. 2 Dynamical mapping of the laser subject to both optical injection and optical feedback in terms of ξf and τf at (ξi, fi) = (0.4, 40 GHz). Red regions, period-one dynamics (P1); Blue regions, period-two dynamics (P2); Gray regions, quasi-periodic dynamics (QP); Black regions, chaos and other instabilities (C).
Fig. 3
Fig. 3 (a) Microwave frequency f0 in terms of ξf at τf = 0.95 ps (black squares), 9.5 ps (red circles), 23.5 ps (blue up-triangles), and 30.51 ps (green down-triangles), respectively. (b) Shifted laser cavity resonance frequency (open symbols) and lower oscillation sideband frequency (closed symbols) in terms of ξf at τf = 0.95 ps (black squares) and 23.5 ps (blue up-triangles), respectively. The y axes of (b) are relative to the free-running frequency of the laser. The injection condition is kept at (ξi, fi) = (0.4, 40 GHz).
Fig. 4
Fig. 4 (a) Microwave frequency f0 in terms of τf at ξf = 0.1 (black squares), 0.2 (red circles), 0.3 (blue up-triangles), and 0.4 (green down-triangles), respectively. (b) Shifted laser cavity resonance frequency (open symbols) and lower oscillation sideband frequency (closed symbols) in terms of τf at ξf = 0.1 (black squares) and 0.4 (green down-triangles), respectively. The y axes of (b) are relative to the free-running frequency of the laser. The injection condition is kept at (ξi, fi) = (0.4, 40 GHz).
Fig. 5
Fig. 5 (a) Feedback loop frequency fl (black curve) and abrupt microwave frequency enhancement Δf0 at ξf =0.1 (black squares), 0.2 (red circles), 0.3 (blue up-triangles), and 0.4 (green down-triangles), respectively, in terms of τf. (b) Reciprocal of T1 (red squares) and frequency difference between the lower P1 oscillation sideband and the free-running laser oscillation (black curve) in terms of ξf.
Fig. 6
Fig. 6 Mappings of microwave frequency f0 in terms of ξf and τf at (ξi, fi) = (0.4, 40 GHz). Each number labeled in the figure indicates the microwave frequency, in GHz, along the boundary between two different gray-scaled regions.
Fig. 7
Fig. 7 Mappings of microwave frequency f0 (a) in terms of θ and ξf at τf = 23.5 ps and (b) in terms of θ and τf at ξf = 0.4, respectively, when (ξi, fi) = (0.4, 40 GHz). The white regions present dynamical states other than the P1 dynamics. Each number labeled in (a) indicates the microwave frequency, in GHz, along the boundary between two different gray-scaled regions.
Fig. 8
Fig. 8 Microwave linewdith Δν (black symbols) and phase noise variance (red symbols) in terms of ξf at (a) τf = 0.95 ps and (b) τf = 23.5 ps, respectively, when (ξi, fi) = (0.4, 40 GHz).
Fig. 9
Fig. 9 Microwave linewdith Δν (black symbols) and phase noise variance (red symbols) in terms of τf at (a) ξf =0.1 and (b) ξf =0.4, respectively, when (ξi, fi) = (0.4, 40 GHz).
Fig. 10
Fig. 10 Mappings of phase noise variance in terms of ξf and τf at (ξi, fi) = (0.4, 40 GHz). The log-scaled values of the phase noise variance are presented.
Fig. 11
Fig. 11 Mappings of phase noise variance (a) in terms of θ and ξf at τf = 23.5 ps and (b) in terms of θ and τf at ξf = 0.4, respectively, when (ξi, fi) = (0.4, 40 GHz). The white regions present dynamical states other than the P1 dynamics. The log-scaled values of the phase noise variance are presented.

Equations (4)

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d a d t = 1 2 [ γ c γ n γ s J ˜ n ˜ γ p ( 2 a + a 2 ) ] ( 1 + a ) + ξ i γ c cos ( Ω i t + ϕ ) + ξ f γ c [ 1 + a ( t τ f ) ] cos [ ϕ ( t τ f ) ϕ ( t ) + θ ] + F a
d ϕ d t = b 2 [ γ c γ n γ s J ˜ n ˜ γ p ( 2 a + a 2 ) ] ξ i γ c 1 + a sin ( Ω i t + ϕ ) + ξ f γ c 1 + a ( t τ f ) 1 + a ( t ) sin [ ϕ ( t τ f ) ϕ ( t ) + θ ] + F ϕ 1 + a
d n ˜ d t = γ s n ˜ γ n ( 1 + a ) 2 n ˜ γ s J ˜ ( 2 a + a 2 ) + γ s γ p γ c J ˜ ( 2 a + a 2 ) ( 1 + a ) 2
ω s = b 2 [ γ c γ n γ s J ˜ n ˜ γ p ( 2 a + a 2 ) ]

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