1. Introduction

By undamping the relaxation resonance of a semiconductor laser through continuous-wave optical injection at a Hopf bifurcation point, period-one (P1) nonlinear dynamics can be invoked [1, 2]. While the optical injection regenerates, oscillation sidebands that are equally separated from the regeneration by an oscillation frequency sharply emerge. Attributed to the cavity resonance red-shift induced by the optical injection, the lower oscillation sideband 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. This gives rise to two highly dominant frequency components separated by the P1 oscillation frequency in the optical spectrum. Because of these unique characteristics, the P1 dynamics have attracted much research interest not only for fundamental understandings of nonlinear dynamics and laser physics [3–9], but also for various technological applications in photonics and microwaves [10–16].

By taking advantage of the two-tone characteristic, the P1 dynamics have been extensively investigated for photonic microwave generation [17–26]. Simply adjusting the power and frequency of the optical injection can continuously tune such generated microwaves from a few to tens or even hundreds of gigahertz [10, 15, 16, 24], an all-optical scheme without suffering from limited electronic bandwidths [27–29]. The two-tone characteristic of the P1 dynamics also manifests a feature of optical single-sideband modulation, which is highly preferred for fiber distribution in radio-over-fiber links to mitigate the microwave power fading effect [30]. However, the spontaneous emission noise of the injected laser deteriorates the spectral purity of the generated microwaves, leading to a considerably broad 3-dB microwave linewidth, typically on the order of 1 to 10 MHz [26]. In addition, fluctuations in the power and frequency of the optical injection relative to those of the injected laser lead to significant microwave frequency jitters, typically on the order of 100 MHz [9]. These characteristics of the P1 microwave oscillations limit the scope of their practical applications.

To improve the microwave spectral purity and stability of the P1 dynamics, a few photonic microwave stabilization approaches have been proposed. By directly modulating the injected laser at the generated microwave frequency, Simpson et al. demonstrated [17] that the generated microwaves could be locked to an electronic microwave reference. While the microwave linewidth was reduced below 1 kHz, the highest locked microwave frequency was restricted to about 17 GHz due to the limited laser response to direct modulation. To exclude the need of an electronic microwave reference, Chan et al. suggested [19] the use of the optoelectronic feedback of the P1 dynamics as the microwave reference. A reduced linewidth of 1 kHz was obtained for microwaves up to 23 GHz. However, a photodetector, an electronic microwave amplifier, and an electronic microwave attenuator, which operated at the generated microwave frequency, were necessary in the feedback loop. As evidently noted, both approaches become more and more difficult or expensive to implement for increasingly high-frequency microwave generation. To bypass the electronic bandwidth restriction, the optical feedback of the P1 dynamics was lately proposed to work as the self-reference for stabilization of the generated microwaves up to 45 GHz [24–26]. However, the microwave linewidth was observed to reduce only by two orders of magnitude, below 50 kHz, owing to the significant microwave frequency jitters that are inherent in either the P1 dynamics scheme for generation [9] or the optical feedback approach for stabilization [26].

In this study, an approach based on optical modulation sideband injection locking is proposed for photonic microwave stabilization of the P1 dynamics. Driving an optical phase modulator by a high-purity electronic microwave reference generates a comb of highly correlated modulation sidebands that are offset from an optical carrier by integral multiples of the reference frequency. Injecting this entire comb-like optical signal into a semiconductor laser excites a P1 dynamical state with key features that is closely similar to the one invoked by a continuous-wave optical signal under the same injection condition. At the same time, a harmonic of the modulation sideband comb, which appears lower than the optical carrier by the P1 oscillation frequency and which possesses an adequate power, injection-locks the lower oscillation sideband of the P1 dynamical state. Since the optical carrier and the harmonic are highly correlated, the regeneration of the optical carrier and the lower oscillation sideband in the P1 dynamical state become phase-locked to each other afterwards. As a result, the poor microwave spectral purity of the P1 dynamical state is improved to a level close to that of the electronic microwave reference. Because of the frequency multiplication nature in yielding the modulation sideband comb, only an electronic microwave reference at a subharmonic, such as the tenth or higher, of the generated microwave frequency is necessary. Therefore, by taking advantage of this stabilization approach, the high-frequency capability requirement of devices and operation is considerably relaxed, while the microwave stability is significantly improved to a much higher level. Following this introduction, the experimental setup is presented in Section 2. Results and analyses are reported in Section 3. Finally, discussion and conclusion are made in Section 4.

2. Experimental setup

Figure 1 presents a schematic of the experimental apparatus using typical single-mode distributed-feedback semiconductor lasers in a master-slave configuration. The slave laser (Furukawa FRL15DCW5-A81) is current-biased at about 6.15 times its 13-mA threshold and temperature-stabilized at 25° C. Under the free-running condition, the slave laser oscillates at 193.33 THz with a power of 10.83 mW at its fiber-pigtail output and with a relaxation resonance frequency of 12.1 GHz. The output of the master laser (Lucent D2525P33) is directed toward the slave laser through a circulator. To excite the P1 dynamics, the frequency of the optical injection is detuned, through adjusting either the temperature or the bias current of the master laser, by f_i from the free-running frequency of the slave laser. In addition, the power of the optical injection is varied using an attenuator and a fiber amplifier, and is measured at the output port of the circulator connected to the slave laser. To indicate the injection strength received by the slave laser, an injection ratio ξ_i, defined as the square root of the power ratio between the optical injection and the free-running slave laser, is used. Note that the power of the optical injection is the composite power of its all spectral components no matter whether it is a continuous-wave or comb-like optical signal used in the following demonstrations. A polarization controller aligns the polarization of the optical injection with that of the slave laser to maximize the injection efficiency. The spectral features of the slave laser output are displayed on an optical spectrum analyzer (Advantest Q8384), and also on a microwave spectrum analyzer (Agilent 8564EC) following a 50-GHz photodiode (u2t Photonics XPDV2120R).

 

Fig. 1 Schematic of the experimental apparatus. ML, master laser; SL, slave laser; PC, polarization controller; PM, phase modulator; MA, microwave amplifier; FA, fiber amplifier; ATT, attenuator; C, circulator; OSA, optical spectrum analyzer; MSA, microwave spectrum analyzer; PD, photodiode.

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To stabilize the P1 dynamics, a 10-GHz optical phase modulator (EOspace PM-0K5-10) is driven by an electronic microwave reference (Agilent E8257D) at a frequency f_m to yield a comb of modulation sidebands which are offset from the master laser by integral multiples of f_m. The spectra of the microwave reference and the resulting comb-like optical signal are shown in Figs. 2(a) and 2(b), respectively, when f_m = 4 GHz. The microwave reference has a 3-dB linewidth of below 1 Hz, which is the highest resolution bandwidth of the microwave spectrum analyzer, as shown in the inset of Fig. 2(a). A comb of only about 2 modulation sidebands on either frequency side is obtained, as shown in Fig. 2(b). To generate a broader modulation sideband comb with greater power in the higher-order harmonics, an electronic microwave amplifier (Picosecond Pulse Labs 5882) working at its saturation regime is used to nonlinearly distort the microwave reference before driving the optical phase modulator. The microwave amplifier used in this study provides a gain of 16 dB, has a saturation output power of 12 dBm at its −1-dB compression point, and allows a maximum input power of 16 dBm. Therefore, by sending a microwave reference with a power of 12 dBm or higher through the microwave amplifier, it is strongly distorted while only slightly amplified. The spectra of such a distorted microwave reference and the resulting comb-like optical signal are shown in Figs. 2(c) and 2(d), respectively, when f_m = 4 GHz. A comb of more than 10 modulation sidebands on either frequency side is generated accordingly. As will be demonstrated, this helps to stabilize the P1 dynamics by using a microwave reference at a frequency much lower than the generated microwave frequency. For comparison, a continuous-wave optical signal of the same optical frequency is also presented as the gray curve in Fig. 2(d).

 

Fig. 2 (a)(b) Spectra of the non-distorted microwave reference and the resulting comb-like optical signal, respectively. (c)(d) Spectra of the distorted microwave reference and the resulting comb-like optical signal, respectively. The inset of (a) shows a close-up of the spectrum, centering at 4 GHz, using the highest resolution bandwidth of 1 Hz. The spectrum of a continuous-wave optical signal (gray curve) is also shown in (d). The x-axes of the optical spectra are relative to the free-running frequency of the slave laser. The microwave reference frequency is fixed at f_m = 4 GHz for each case.

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3. Results and analyses

3.1. P1 dynamics

First consider the situation when the slave laser is subject to continuous-wave optical injection only. By injecting the continuous-wave optical signal, the gray curve shown in Fig. 2(d), into the slave laser at (ξ_i, f_i) = (1.28, 31 GHz), a P1 dynamical state is excited, as Fig. 3(a) presents. A regeneration of the optical injection appears at the offset frequency of 31 GHz, which results from the injection pulling effect [31]. In addition, oscillation sidebands, which are equally separated from the regeneration by an oscillation frequency of f₀ = 40 GHz, sharply emerge through undamping the relaxation resonance of the slave laser. Since the optical injection reduces the necessary gain for the injected slave laser, the laser cavity resonance red-shifts through the antiguidance effect [8]. Accordingly, the lower oscillation sideband is resonantly enhanced as opposed to the upper one. As a result, the lower oscillation sideband is not only 23 dB stronger than the upper one, but also has a power close to the regeneration, only 5 dB weaker. Effectively, the optically injected laser system at the P1 dynamics functions as a two-tone optical oscillator, leading to a feature of optical single-sideband modulation. Various P1 dynamical states similar to the one shown in Fig. 3(a) can be excited over a wide range of ξ_i and f_i [1, 2, 7, 8, 20]. The feasibility of exciting various P1 dynamical states over a broad range of the optical injection condition provides a possibility to dynamically reconfigure the same laser system through a simple all-optical adjustment for different operating requirements in practical applications. For example, f₀ can be enhanced by increasing ξ_i or/and f_i, giving rise to a broadly and continuously tunable f₀ from a few up to tens or even hundreds of gigahertz [10, 15, 16, 24].

 

Fig. 3 (a) Optical and (b) microwave spectra of the P1 dynamics when the slave laser is subject to the continuous-wave optical injection shown in Fig. 2(d) at (ξ_i, f_i) = (1.28, 31 GHz). The x-axis of the optical spectrum is relative to the free-running frequency of the slave laser. The microwave spectrum centers at 40 GHz with a resolution bandwidth of 1 MHz. The trace of the microwave frequency jitters (gray curve) is also shown in (b) for an observation period of 100 seconds.

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The beating between the spectral components shown in Fig. 3(a) at the photodiode generates a microwave signal jittering around 40 GHz, as Fig. 3(b) presents. Since the relaxation resonance frequency of the free-running slave laser is about 12.1 GHz, this result demonstrates the capability of the P1 dynamic scheme for very high-frequency microwave generation without suffering from the limited laser response. As noted, the generated microwave signal does not result from a simple optical heterodyne between the optical injection and the free-running slave laser, but from a nonlinear dynamical process within the optically injected slave laser. By capturing the microwave signal at a time instant before it jitters to the next frequency, its 3-dB linewidth of about 1.7 MHz is estimated by fitting the spectrum with a Lorentzian curve. This considerably broad microwave linewidth results mostly from the spontaneous emission noise of the injected slave laser [26], assuming the optical injection is highly coherent when the master laser is current-biased well above its threshold. This is verified by the fact that the optical linewidth of the slave laser used in this study is on the order of megahertz. A trace of the microwave frequency jitters, about 151-MHz wide, for an observation period of about 100 seconds is presented as the gray curve in Fig. 3(b). Such significant frequency jitters arise from fluctuations in the power and frequency of the optical injection relative to those of the slave laser due to slight variations in the currents and temperatures of both lasers [9]. These poor microwave characteristics need to be improved before the P1 dynamics scheme is made more practically feasible for applications that require high-purity and high-frequency microwave generation.

3.2. P1 dynamics stabilization

To improve the microwave spectral purity and stability of the P1 dynamical state shown in Fig. 3(b), establishing a phase-locking between the regeneration and the lower oscillation sideband shown in Fig. 3(a) is necessary. This can be achieved by injection-locking them to two highly correlated frequency components, respectively, of an optical signal. The comb-like optical signal shown in Fig. 2(d) can be used to work as such an injection locking signal. Figure 4 demonstrates the optical and microwave spectra by injecting the entire comb-like optical signal shown in Fig. 2(d) into the slave laser at the same (ξ_i, f_i) = (1.28, 31 GHz). As presented in Fig. 4(a), except the tightly spaced side peaks resulting from the regeneration of the modulation sideband comb, the optical carrier of the comb-like optical signal excites a P1 dynamical state with key features that is closely similar to the one shown in Fig. 3(a). A regeneration of the optical carrier and a lower oscillation sideband appear, which are separated by f₀ = 40 GHz and which dominate the optical spectrum. At the same time, the tenth harmonic of the lower modulation sideband comb, which is 40-GHz lower than the optical carrier, injection-locks the lower oscillation sideband of the P1 dynamical state. Since the optical carrier and the tenth harmonic are highly correlated, the regeneration of the optical carrier and the lower oscillation sideband in the P1 dynamical state now become phase-locked to each other. Consequently, as presented in Fig. 4(b), a stable microwave generation at 40 GHz with a 3-dB linewidth of below 1 Hz, same as the microwave reference, is achieved. No microwave frequency jitters are observed over a time period of more than 30 minutes under study, while only slight microwave power fluctuations, within 1 dB, are observed. This result also verifies that the regeneration and the lower oscillation sideband of the P1 dynamical state are well injection-locked by the optical carrier and the tenth harmonic, respectively, of the comb-like optical signal, which are highly correlated owing to the inherent nature of optical modulation.

 

Fig. 4 (a) Optical and (b) microwave spectra of the P1 dynamics when the slave laser is subject to the comb-like optical injection shown in Fig. 2(d) at (ξ_i, f_i) = (1.28, 31 GHz). The x-axis of the optical spectrum is relative to the free-running frequency of the slave laser. The microwave spectrum centers at 40 GHz with a resolution bandwidth of 1 Hz. The microwave reference frequency is fixed at f_m = 4 GHz.

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To further demonstrate the stability of the generated 40-GHz microwave signal, its single-sideband (SSB) phase noise, estimated as the ratio of the power at a non-zero frequency offset to that at the zero, is demonstrated in Fig. 5. To faithfully capture the phase noise at extremely low levels likely observed in this study, another microwave spectrum analyzer (Agilent N9030A PXA) is used for the measurement. The phase noise level of the stabilized microwave generation is highly comparable to the ones that are possibly the lowest found in the literature [32], where photonic microwave generation schemes based on other multiwavelength light sources were investigated. For example, at the offset frequency of 100 kHz, the phase noise is about −91 dBc/Hz shown in Fig. 5, around −94 dBc/Hz for actively mode-locked semiconductor lasers [33], and approximately −100 dBc/Hz for optical heterodyning of two semiconductor lasers [34]. To investigate whether the generated 40-GHz microwave signal can be further stabilized by reducing its phase noise to a lower level, the phase noise of the 4-GHz microwave reference itself is also presented in Fig. 5. In addition, since the SSB phase noise scales with the square of the frequency multiplication N [35], the phase noise of the microwave reference scaled up by 10[log₁₀(N²)] = 20 dB for N = 10 in this demonstration is shown as well for fair comparison. Excess phase noise of approximately 9 to 25 dB is observed over the offset frequency range under study. This is attributed to the non-perfect locking between the tenth harmonic and the lower oscillation sideband, resulting from the weakness of the tenth harmonic. Limited by the power of the devices used in this study, the maximum attainable power of the tenth harmonic is only about −34 dBm, which is about 43-dB weaker than the optical carrier, as shown in Fig. 2(d). As will be deduced from the results shown in Fig. 6, the higher the power of a harmonic is, the lower the phase noise of a microwave generation is. For example, if the power of the tenth harmonic can be increased by 15 dB, it is possible for the excess phase noise to considerably reduce down to about 1 to 15 dB over the offset frequency range under consideration. This suggests that the proposed stabilization approach can effectively and considerably improve the poor microwave stability of the P1 dynamical state of f₀ = 40 GHz up to an extent highly close to the stability of the microwave reference at f_m = 4 GHz, where f_m = f₀/N and N = 10 in this demonstration. Based on the scaled phase noise presented in Fig. 5, a phase noise level down to below −100 dBc/Hz at the 100-kHz offset frequency is therefore highly possible for the generated microwave signal to achieve if the tenth harmonic has a sufficient power.

 

Fig. 5 Single-sideband (SSB) phase noise as a function of microwave offset frequency for the generated 40-GHz microwave signal (black solid curve), the 4-GHz microwave reference (red solid curve), and the 4-GHz microwave reference scaled by N = 10 (red dotted curve).

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Fig. 6 (a) Microwave spectrum of the generated 20-GHz signal, centering at 20 GHz with a resolution bandwidth of 1 Hz, when f_m = 4 GHz and N = 5. (b) Phase noise in terms of offset frequency for the generated 20-GHz signal in (a) (black solid curve), the 4-GHz reference (red solid curve), and the 4-GHz reference scaled by N = 5 (red dotted curve). (c) Phase noise variance in terms of N for generated microwave signals (blue symbols) and scaled microwave references (red symbols) when f_m = 4 GHz. (d) Microwave power (black symbols) and SCR (white symbols) in terms of N when f_m = 4 GHz. The corresponding values of f₀ are also marked in the upper x-axes of (c) and (d).

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The stabilization of the generated 40-GHz microwave signal fails when the power of the tenth harmonic is reduced by less than 1 dB from −34 dBm. This suggests that −34 dBm is approximately the lowest required power for a harmonic of a comb-like optical signal to stably lock the lower oscillation sideband of a P1 dynamical state under study. Therefore, for the comb-like optical signal of f_m = 4 GHz shown in Fig. 2(d), if a harmonic of N = 15, for example, emerges with an adequate power, a P1 dynamical state of f₀ = 60 GHz can be stabilized. On the other hand, for the P1 dynamical state of f₀ = 40 GHz shown in Fig. 3, a comb-like optical signal of f_m = 2.5 GHz, for example, can be used for stabilization if a harmonic of N = 16 emerges with a power sufficient to stably lock the lower oscillation sideband. Even though the device bandwidth limits such demonstrations in the this study, they can be practically achieved, for example, by using an optical phase modulator of the same speed but with a slower roll-off at high frequencies, or one of a higher speed, such as 40 GHz, which is now commercially available.

The discussion on Fig. 4(b) points out that no microwave frequency jitters are observed under study. This indicates that the proposed stabilization approach works even if f₀ in Fig. 3 shifts around 40 GHz, resulting from the fluctuation of the injection condition, when f_m = 4 GHz is fixed. For the tenth harmonic at the power level of −34 dBm, the lower oscillation sideband is similarly locked as in Fig. 4 if f₀ shifts within a locking range of 300 MHz around 40 GHz. This 300-MHz locking range corresponds to a change of about 0.03 around ξ_i = 1.28 or a change of about 1 GHz around f_i = 31 GHz. On the other hand, the approach also works if f_m shifts around 4 GHz when f₀ = 40 GHz is kept. The tenth harmonic at the power level of −34 dBm still locks the lower oscillation sideband if f_m shifts within a locking range of 20 MHz around 4 GHz, which corresponds to a change of about 200 MHz around 40 GHz for the tenth harmonic. These locking ranges become broader if the power of the harmonic is higher, as will be suggested by the results presented in Fig. 6.

As addressed above, by adopting a different harmonic order N, a different microwave reference frequency f_m, or their combination, a P1 dynamical state of different f₀ can be stabilized through the proposed approach. This enables photonic microwave generation of not only broadband tunability but also high spectral purity using the P1 dynamics scheme. For further demonstration, two different scenarios are considered as follows. First, for a comb-like optical signal of f_m = 4 GHz, such as the one shown in Fig. 2(d), a harmonic of different N can be used to stabilize a P1 dynamical state of f₀ = 4 × N. For example, a P1 dynamical state of f₀ = 20 GHz is excited at (ξ_i, f_i) = (0.95, 3 GHz), and is stabilized using a harmonic of N = 5 to injection-lock its lower oscillation sideband. A stabilized microwave generation at 20 GHz with a 3-dB linewidth of below 1 Hz is therefore obtained, as Fig. 6(a) presents. Compared with the result shown in Fig. 5, its SSB phase noise is, on average, about 16-dB lower across the entire offset frequency range, as Fig. 6(b) demonstrates. Theoretically, the phase noise of the fifth harmonic used for injection-locking, which is scaled up from the phase noise of the 4-GHz microwave reference by 10[log₁₀(N²)] ≈ 14 dB for N = 5, is 6-dB lower than that of the tenth harmonic. Therefore, this lower phase noise level of the fifth harmonic contributes approximately 6 dB to such 16-dB reduction. The remaining 10-dB reduction results from a better locking between the fifth harmonic and the lower oscillation sideband. This is verified by the fact that the fifth harmonic has a power of about −19 dBm and is therefore 15 dB stronger than the tenth harmonic considered in Fig. 5. Consequently, as observed in Fig. 6(b), the excess phase noise is significantly lower, which is about 1 to 17 dB over the offset frequency under study, and the phase noise at the 100-kHz offset frequency is only about −107 dBc/Hz. Since the fifth harmonic is 15-dB stronger than the tenth harmonic discussed in Fig. 5, the aforementioned locking ranges are broadened accordingly. Injection-locking is assured if f₀ shifts within a locking range of 1 GHz around 20 GHz, which corresponds to a change of about 0.04 around ξ_i = 0.95 or a change of about 2.5 GHz around f_i = 3 GHz. In addition, it is maintained if f_m shifts within a locking range of 100 MHz around 4 GHz, which corresponds to a change of about 500 MHz around 20 GHz for the fifth harmonic.

Similar stabilized microwave generations at f₀ = 24, 28, 30, 36, and 40 GHz using different P1 dynamical states are achieved for N = 6 to 10, respectively. The phase noise variance of each stabilized microwave generation is shown in Fig. 6(c), which is estimated by integrating its SSB phase noise from the frequency offset of 100 Hz to 1 MHz, and is compared with that of the 4-GHz microwave reference scaled up by its corresponding N. For ease of comparison, both N and the corresponding f₀ are marked in the lower and upper x-axes, respectively. The phase noise variance of the generated microwave signals generally enhances with N, which is attributed to the increased phase noise and decreased power of a harmonic used for injection-locking when N increases. It is about one to two orders of magnitude higher than that of the scaled microwave reference. Such excess phase noise can be reduced if the power of the harmonic used for injection-locking is enhanced. These phase noise performances demonstrate that each noisy P1 dynamical state is well injection-locked to the highly correlated comb-like optical signal and that a high level of microwave spectral purity is achieved accordingly. As shown in Fig. 6(d), each P1 dynamical state of different f₀ demonstrated in Fig. 6(c) is so chosen that the power ratio of the lower oscillation sideband to the regeneration of the optical carrier, referred to as the sideband-to-carrier ratio (SCR), is about the same, around −0.5 dB. This leads to a similar microwave power level over a broad frequency range of microwave generation, as also shown in Fig. 6(d), if the same optical power is received by the photodiode. Note that the microwave power can be maximized for any received optical power if SCR approaches zero.

For the second scenario, considering a harmonic of N = 10, a comb-like optical signal of different f_m can be used to stabilize a P1 dynamical state of f₀ = 10 × f_m. For example, a P1 dynamical state of f₀ = 20 GHz is excited at (ξ_i, f_i) = (0.78, 4 GHz). It is stabilized using a comb-like optical signal of f_m = 2 GHz with the tenth harmonic injection-locking its lower oscillation sideband. A stabilized microwave generation at 20 GHz with a 3-dB linewidth of below 1 Hz, as shown in Fig. 7(a), is therefore achieved. Figure 7(b) presents that, on average, its SSB phase noise is around 14-dB lower than that of the generated 40-GHz microwave signal shown in Fig. 5 across the entire offset frequency range. Since the phase noise of the 2-GHz microwave reference is, on average, about 7-dB lower than that of the 4-GHz microwave reference, it contributes 7 dB to such 14-dB reduction. A better locking between the tenth harmonic and the lower oscillation sideband for the generated 20-GHz microwave signal leads to the remaining 7-dB reduction. This is attributed to the fact that the tenth harmonic of f_m = 2 GHz generated by the experimental setup in this study has a power of about −26 dBm and is therefore 8-dB stronger than that of f_m = 4 GHz considered in Fig. 5. Accordingly, as observed in Fig. 7(b), the excess phase noise ranges from 4 to 21 dB over the offset frequency under investigation, and the phase noise at the 100-kHz offset frequency is only about −104 dBc/Hz.

 

Fig. 7 (a) Microwave spectrum of the generated 20-GHz signal, centering at 20 GHz with a resolution bandwidth of 1 Hz, when f_m = 2 GHz and N = 10. (b) Phase noise in terms of offset frequency for the generated 20-GHz signal in (a) (black solid curve), the 2-GHz reference (red solid curve), and the 2-GHz reference scaled by N = 10 (red dotted curve). (c) Phase noise variance in terms of f_m for generated microwave signals (blue symbols) and scaled microwave references (red symbols) when N = 10. (d) Microwave power (black symbols) and SCR (white symbols) in terms of f_m when N = 10. The corresponding values of f₀ are also marked in the upper x-axes of (c) and (d).

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Similar stabilized microwave generations at f₀ = 25, 30, 35, and 40 GHz using different P1 dynamical states are obtained for f_m = 2.5, 3, 3.5, and 4 GHz, respectively. The phase noise variance of each stabilized microwave generation is shown in Fig. 7(c), and is compared with that of the f_m microwave reference scaled up by N = 10. For ease of comparison, both f_m and the corresponding f₀ are marked in the lower and upper x-axes, respectively. The phase noise variance of the generated microwave signals generally enhances with f_m, which is attributed to the increased phase noise and decreased power of the tenth harmonic used for injection-locking when f_m increases. It is about one to two orders of magnitude higher than that of the scaled microwave reference. Such excess phase noise can be reduced if the power of the harmonic used for injection-locking is enhanced. These phase noise performances again demonstrate that each noisy P1 dynamical state is well injection-locked to the highly correlated comb-like optical signal and that a high level of microwave spectral purity is achieved accordingly. The different P1 dynamical states studied in Fig. 7(c) are chosen so that their SCR values are again around −0.5 dB, as Fig. 7(d) shows, giving rise to a microwave power level similar to that presented in Fig. 6(d). Hence, the results presented in Figs. 6 and 7 demonstrate that highly stable and broadly tunable photonic microwave generation with constant power output is feasible using the P1 dynamics scheme together with the proposed stabilization approach. Note that even though the stabilization approach is investigated using P1 dynamical states with an optical single-sideband feature similar to that shown in Fig. 3(a) where the lower sideband is much stronger than the upper one, it also works well for those with different power ratios between the two sidebands.

4. Discussion and conclusion

A similar stabilization approach based on the optical modulation sideband injection locking has been applied to the microwave generation based on the optical heterodyne scheme using two independent lasers to interfere on a photodiode [34, 36, 37]. As opposed to the P1 dynamics scheme addressed in this study, there require strict operating conditions in microwave stabilization for the optical heterodyne scheme. First, well-matched optical path lengths to the photodiode are necessary for stable interference between the two independent lasers. Second, differences in acoustic noise from the environment coupling into the two independent optical paths are required to reduce for phase noise suppression at low offset frequencies. Third, the optical carrier of the comb-like optical signal needs to be considerably suppressed or even completely removed before injection to prevent laser instability. The P1 dynamics scheme under study therefore provides a photonic microwave generation system of simpler configuration and operation.

In summary, improving the microwave phase quality of P1 nonlinear dynamics in semiconductor lasers is important to take advantage of the dynamics for stabilized photonic microwave generation. This study investigates a photonic microwave stabilization approach based on optical modulation sideband injection locking for such a purpose. High spectral purity, a 3-dB linewidth of below 1 Hz, for such photonic microwave generation up to 40 GHz is demonstrated. The highest demonstrable frequency is mainly restricted by the bandwidth of the devices used in this study, not by the proposed microwave generation and stabilization scheme. Stabilized microwave generation at a higher frequency, such as 100 GHz or more, is feasible. By taking advantage of frequency multiplication in yielding the optical modulation sidebands, only an electronic microwave reference at a subharmonic, such as the tenth or higher, of the generated microwave frequency is required for the photonic microwave stabilization.