1. Introduction

Microwave photonics represents an exciting field, which provides original solutions to overcome imminent bottlenecks in electronic circuits [1]. Over the past decades, the photonic generation of microwave signals has been widely applied in various fields, such as radio-over-fiber (ROF), THz spectroscopy, photonic-based remote sensing, and all-optical information and signal processing [2–7]. So far, several photonic microwave generation techniques have been proposed and demonstrated, including direct modulation [8,9], external modulation [10,11], optoelectronic oscillators (OEOs) [12,13], optical heterodyning [14,15], and period-one (P1) oscillations [16–19]. In particular, the approach using P1 oscillations of an optically injected semiconductor laser (OISL) to generate photonic microwave signals has attracted intense attention, since it is only involved with all-optical components, operates nearly in the single-sideband spectrum, and allows for broadband and tuning microwave frequencies. In the OISL system, the P1 dynamics are achieved when a stably locked semiconductor laser undergoes a Hopf bifurcation and exhibits the two dominant frequencies in the optical spectrum, i.e., the red-shifted cavity frequency and the regenerated frequency from the optical injection. Their beating would generate a microwave signal at the output of the photodetector. By adjusting the injection parameters, the photonic microwave signal can reach up to the frequency of ten times the relaxation oscillation (RO) frequency of a solitary semiconductor laser. Recently, a series of semiconductor lasers have been employed to achieve the photonic generation of microwave signals, such as distributed feedback (DFB) lasers [20,21], quantum dot (QD) lasers [22,23], and vertical-cavity surface-emitting lasers (VCSELs) [24–26]. In spite of outstanding progress in the photonic microwave signal generation based on P1 oscillations, most schemes are built by following a master-slave configuration, which inevitably increases the complexity of the system structure. In such a configuration, the field noise generated from the master laser markedly increases the microwave central linewidth, which is undesired in practical applications [27]. Moreover, fluctuations in the power and frequency of the optical injection relative to those of the injected laser cause significant microwave frequency jitters, typically on the order of 100 MHz [17].

An alternative approach to generating photonic microwave signals is involved with a solitary dual-mode laser. It not only avoids the effect of external injection parameters on the photonic microwave signal, but also minimizes the size of the microwave signal source and thus facilitates the integration of microwave photons in practical applications. Sun et al. experimentally demonstrated that a single-longitudinal-mode dual-wavelength DFB fiber laser with a wavelength spacing of 0.312 nm supports the generation of a microwave signal at 38.67 GHz by beating the two-lasing wavelength [28]. However, it has a drawback in the tuning ability because of the fixed frequency spacing between the two lasing modes. Zhou et al. numerically demonstrated a 30 GHz microwave signal in a parallelly-polarized optically injected VCSEL with the injection wavelength close to the suppressed polarized mode, in which the generated microwave frequency relies on the frequency difference between the dominant mode and the suppressed orthogonal mode [29]. More interestingly, such a frequency difference, the so-called birefringence splitting, can be enlarged (≥ 250 GHz) by using several technologies, such as mechanically applied in-plane anisotropic strain [30], asymmetric current-induced heating [31], and integrated surface grating [32]. With such features, they stick out as appealing candidates for microwave-photonic applications. Nevertheless, the suppressed polarization mode hinders the generation of photonic microwave signals for a conventional VCSEL. Recently, a novel type of spintronics devices, i.e., the optically pumped spin-polarized VCSEL (spin-VCSEL), has been proposed and demonstrated to generate photonic microwave signals due to the fact that it can yield high-frequency continuous birefringence-induced oscillation without introducing any external perturbation. In such a laser, the polarization mode of the oscillation is dominated by the competition between the residual birefringence intrinsic to the semiconductor structure and the circular dichroism gain emerging from the spin imbalance in the active medium [33,34]. Torre et al. experimentally investigated a tunable P1 oscillation whose frequency can be increased from 8.6 GHz to 11 GHz, when the pump polarization ellipticity is changed in an optically pumped quantum well (QW) spin-VCSEL [35]. Afterwards, we numerically demonstrated the generation of the photonic microwave in a wide parameter range. However, our results display that the generated microwave signals with a dominant linewidth of about 3 MHz have a widely tunable frequency (from several gigahertz to hundreds of gigahertz) [36]. We also preliminary demonstrated that microwave linewidth can be reduced by introducing the single polarization preserved optical feedback.

In this paper, we extend our earlier work reported in Ref. [36]. First, we introduce an axial magnetic field in an optically pumped spin-VCSEL and discuss the effect of the value and signal of the magnetic field on the microwave frequency and linewidth, which can provide a way to adjust the frequency of P1 microwave when the birefringence split is fixed. Second, we systematically investigate the role of optical feedback in microwave linewidth narrowing. The use of single-loop feedback can reduce the microwave linewidth though increasing the feedback strength and/or delay time, whereas the side peaks in the radio-frequency (RF) spectrum are magnified. By adding the dual-loop feedback, the side peaks can be suppressed by the Vernier effect deploying incommensurate delay times.

2. Scheme and theoretical model

The proposed photonic microwave generation scheme is displayed in Fig. 1(a). The polarization of the pump was controlled with a polarization controller. Under a selected parameter, the spin-VCSEL operates in P1 oscillation. Figure 1(b) shows the optical spectrum emitted by the spin-VCSEL, where the output of the laser is projected in two orthogonally linear polarization modes. We define the frequencies in the optical domain as ν and the frequencies in the electrical domain as f. Afterwards, the beam is coupled with a standard single-mode fiber and then the emitted light is sent on a high-speed photodiode (PD). Before the PD, a polarization controller and polarizer are inserted to project the orthogonal linear polarizations on the same optical axis [see Fig. 1(c)]. In the electrical domain, one can observe (i) a central peak f₁ corresponding to the beat frequency between two projected linear polarizations and (ii) the satellite peak at f₂ corresponding to the beatnote frequencies between the x(y) mode at the order p and the y(x) mode at the order p + 1(p-1). Moreover, the experimental results demonstrated that the generated microwave signals were widened due to the fact that the spontaneous emission noise broadens the linewidth of the optical spectrum [35]. Hence, we introduce optical feedback configurations (owing to their desired properties, i.e., the simple and all-optical structure) to improve the performance of the photonic microwave produced from the spin-VCSEL and a detailed discussion is presented in sections 3-2 and 3-3.

 

Fig. 1. (a) Schematic diagram of the photonic microwave generated from the spin-VCSEL. PC, polarization controller; CIR, optical circulator; OI, optical isolator; PD, photodetector; Pol, polarizer; ESA, electrical spectral analyzer; OSA, optical spectral analyzer; OSC, oscilloscope. The gray lines represent the optical feedback loops. (b) The optical spectrum emitted by the spin-VCSEL. (c) Associated optical spectrum after projection on the same polarization axis using a polarizer. (d) Corresponding RF spectrum.

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Our simulation is carried out by the extended spin-flip model (SFM) to include an axial magnetic field and two polarization-reserved optical feedback configurations, which can be written as [36,37]:

In this model, the intrinsic parameters are described as follows: κ is the optical field decay rate, γ is the carrier recombination rate, α is the linewidth enhancement factor, γ_s is the spin relaxation rate, γ_p is the linear birefringence, and γ_a is the dichroism rate. There are two experimentally controllable parameters, i.e., the total normalized pump intensity $\eta = {\eta _ + } + {\eta _ - }$ and the pump polarization ellipticity $P = ({\eta _ + } - {\eta _ - })/({\eta _ + } + {\eta _ - })$, where η₊ and η_- correspond to the right and left circular polarization components of the pump. The third and four terms in Eq. (1) are the feedback terms, where k_f1,2 denote the feedback strength, τ_1,2 represent the feedback delay time. The central frequency of the VCSEL is indicated as ${f_0} = ({f_x} + {f_y})/2$, where $2\pi {f_x} = 2\pi {f_0} + \alpha {\gamma _a} - {\gamma _p}$ and $2\pi {f_y} = 2\pi {f_0} - \alpha {\gamma _a} + {\gamma _p}$ are separately the frequency of the x and y polarization components. Furthermore, the spontaneous emission noise F_x,y are modeled as [38]:

3. Results and discussion

3.1 Birefringence-induced P1 microwave

Usually, because of the cavity anisotropies, VCSELs operate in one of two orthogonally linearly polarized modes, in which they have different emission frequencies. Following the SFM, the steady-state frequency splitting can be described as $\Delta f = ({\gamma _p} - \alpha {\gamma _a})/\pi $, where the gain saturation is neglected [40]. When the dichroism γ_a and linewidth enhancement factor α are small values, the frequency difference is primarily determined by the linear birefringence in the cavity described by γ_p. In particular, under the spin carrier injection, the laser emits dominantly on a circularly polarized laser mode, which corresponds to a simultaneous emission of both orthogonally linearly polarized laser modes. After the frequency beating, the polarized oscillation (PO) can be observed and the oscillation frequency depends on the frequency splitting (that is, the linear birefringence in the cavity). Figure 2 illustrates the optical spectra and associated RF spectra of the free-running spin-VCSEL. In Fig. 2(a1), we can see that the two modes are split by Δf = 31.77 GHz and generate a microwave signal with an oscillation frequency close to γ_p/π [see Fig. 2(b1)], rather than the RO frequency (5.03 GHz). Under such a scenario, the microwave frequency can be tuned by changing the birefringence. For example, the birefringence splitting can be tuned to 45 GHz via asymmetric heating as previously mentioned, and the optical spectrum is shown in Fig. 2(b1), which results in microwave signals with a frequency of 45 GHz [see Fig. 2(b2)]. When the birefringence splitting is further increased to 98 GHz, e.g., by using monolithically integrating tailored surface grating, a microwave signal with a frequency close to a hundred gigabytes can be achieved, as shown in Fig. 2(c). Interestingly, the birefringence splitting was experimentally demonstrated to be beyond 250 GHz through the mechanically applied in-plane anisotropic strain and thus microwave signals over hundreds of gigabytes can be acquired in a free-running spin-VCSEL. In addition, similar to the conventional semiconductor lasers, the spontaneous emission noise inevitably increases the linewidth of optical sidebands of the spin-VCSEL and thus degrades the performance of microwave signals, such as broad microwave linewidth (∼MHz), as shown in the power spectra of Figs. 2(a2)-(c2).

 

Fig. 2. (a1)-(c1) Optical spectra and (a2)-(c2) RF spectra of a solitary spin-VCSEL, where (a) γ_p = 30π ns^-1, (b) γ_p = 44.5π ns^-1, and (c) γ_p = 97π ns^-1.

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To further study the changing of P1 microwave linewidth Δf₀, we plot the color map of the linewidth in the (γ_s, γ_p) plane in Fig. 3(a), where only the P1 oscillation is calculated, and other dynamics regimes are marked in white. Herein, the P1 microwave frequencies are indicated using white contour lines. As expected, one can see that the microwave frequency is dominated by birefringence γ_p rather than the RO frequency f_r, which is an intrinsic feature of the spin-VCSEL [41]. The microwave linewidth Δf₀ of about 3 MHz occupies a broad region in the P1 oscillation in Fig. 3(a). Interestingly, within the region of P1 oscillations, when the microwave frequency grows, the microwave central linewidth is not significantly broadened. However, we also find that, when the laser parameters are close to the Hopf bifurcation on the left [36], the microwave linewidth enlarges dramatically, varying from 8 MHz to hundreds of megahertz.

 

Fig. 3. Microwave linewidth Δf₀ and microwave frequency f₀ are calculated (a) in the (γ_s, γ_p) plane, where (η, P) = (3, 0.5), and (b) in the (η, P) plane, where (γ_s, γ_p) = (35 ns^-1, 30π ns^-1). The magnetic field is not considered (i.e., Ω_z = 0). The white contour lines represent the microwave frequency f₀.

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In our previous work, we showed the effect of two experimentally controllable parameters (e.g., η and P) on the quality of the P1 microwave signal including the oscillation frequency, amplitude, power, and linewidth [36]. In the present work, we focus on the effect of the axial magnetic field Ω_z on the P1 microwave signal since it can significantly affect the generation of P1 oscillations in an optically pumped spin-VCSEL [37]. For comparison, we first show the scenario of a spin-VCSEL without a magnetic field (i.e., Ω_z = 0) in Fig. 3(b). As can be seen that the region of P1 oscillations is symmetric around P = 0 and the microwave linewidth of 3 MHz occupies a wider range in the (η, P) plane. As expected, a broad microwave central linewidth is observed in the Hopf bifurcation boundary because the spin-VCSEL has just gone through a destabilizing process. Moreover, we want to point out that the η and P are not dominant parameters to increase the microwave frequency in our proposed scheme. After that, we consider the scenario of a spin-VCSEL with an axial magnetic field (i.e., Ω_z = 35 rad·ns^-1), as shown in Fig. 4(a). From this figure, we find that the symmetry of the P1 oscillation region is broken and the P1 region is expanded in the upper half of the maps such that the P1 oscillation can appear in the scenario of P = 0 (corresponding to a conventional VCSEL) when η > 2.6. Nevertheless, we note that the region of broad microwave linewidth emerging in the edge of the upper half of the map is obviously expanded compared with the case of Ω_z = 0 [see Fig. 3(b)]. Outside this range, the microwave linewidth still remains around 3 MHz. In the lower half of the map, the region of P1 oscillation is shrunken sharply in size and shifted to a larger value of |P|. Fortunately, the microwave linewidth is not widened and shows a similar trend to the case of Ω_z = 0. Moreover, we also calculate microwave frequencies, which are marked by white contour lines in Figs. 3(b) and 4(a). From these figures, by increasing either the value of η or |P|, the P1 oscillation frequency is enlarged for both scenarios. Nonetheless, in the upper half of the map, the presence of an axial magnetic field visibly increases the P1 microwave frequency, whereas the microwave frequency is decreased in the lower half of the map compared with the case of Ω_z = 0. Additionally, we also consider the effect of the sign of the magnetic field in Fig. 4(b), where Ω_z = -35 rad·ns^-1. By comparing Fig. 4(a) with Fig. 4(b), one can see that the calculated P1 oscillation maps with opposite signs of Ω_z are just mirror images about P = 0. This result may provide a choice to manipulate the characteristics of the P1 microwave signals generated from an optically pumped spin-VCSEL. For example, the introduction of the magnetic field enables the microwave frequency to be tuned within a certain range when the intrinsic parameters (e.g., γ_p and α) are fixed.

 

Fig. 4. Microwave linewidth Δf and microwave frequency are calculated in the (η, P) plane, where (a) Ω_z = 35 rad·ns^-1 (b) Ω_z = -35 rad·ns^-1. The white contour lines represent the microwave frequency f_o. The other parameters are the same as those in Fig. 3(b).

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In this section, we show the birefringence-induced P1 microwave signals generated from a free-running spin-VCSEL. Compared with the optically injected VCSEL scheme, our proposed scheme has a simple structure and broad tunable microwave frequency from a few gigahertz to hundreds of gigahertz by increasing the linear birefringence. Compared with our previous work [36], here we further discuss the effect of the magnetic field on the P1 microwave. These results demonstrate that the P1 microwave frequency f_o can be adjusted beyond the RO frequency, while it exhibits a broad microwave linewidth Δf₀ and acts as the oscillation phase noise. To improve the quality of microwave signals, optical feedback is introduced to the spin-VCSEL in sections 3-2 and 3-3.

3.2 P1 with single-loop feedback

Microwave linewidth narrowing via single-loop optical feedback is introduced to a spin-VCSEL, where the feedback constructs external cavity modes and leads to the decrease of the microwave linewidth for the central peak along with residual side peaks separated by the reciprocal of the feedback delay time. In the following study, the axial magnetic field Ω_z is set to 0. Initially, only one loop in Fig. 1(a) is switched on. Under this scenario, the feedback parameters (k_f1, τ₁) are indicated as (k_f, τ), while the other loop is kept off and k_f2 is set at zero. Figure 5 displays the RF spectra of the spin-VCSEL emission at different values of k_f, where τ = 5 ns. Here, the intrinsic parameters of the spin-VCSEL are fixed at (γ_s, γ_p) = (35 ns^-1, 30π ns^-1), corresponding to the P1 oscillation frequency at f₀ = 31.77 GHz [see Fig. 2(a2)].

 

Fig. 5. Microwave power spectra of the spin-VCSEL with single-loop feedback. The feedback strength k_f = (a) 0.3 ns^-1, (b) 0.7 ns^-1, (c) 1.1 ns^-1, and (d) 1.5 ns^-1. The feedback delay time is fixed at τ = 5 ns. The intrinsic parameters are fixed at (γ_s, γ_p) = (35 ns^-1, 30π ns^-1).

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In Fig. 5(a) with k_f = 0.3 ns^-1, the spin-VCSEL operates in the P1 oscillation at f₀ = 31.77 GHz. However, compared with Fig. 2(a2), the RF spectrum in Fig. 5(a1) displays an obvious reduction of the central peak at f₀. Quantitatively, the microwave linewidth decreases from the original Δf₀ = 3.14 MHz down to Δf₀ = 215 kHz. Despite the existence of some side peaks equally separated from the central peak by multiples of 1/τ introduced by the optical feedback, these peaks are much smaller than the central peak (over 50 dB), which does not disturb the computing of the microwave central linewidth. We further increase the feedback strength to k_f = 0.7 ns^-1, and the microwave linewidth is further decreased to Δf₀= 75 kHz, as shown in the inset of Fig. 5(b). In Fig. 5(c), as the feedback strength is increased to k_f = 1.1 ns^-1, the microwave linewidth decreases to Δf₀= 30 kHz, which is about two orders of the magnitude smaller than that of the origin microwave linewidth [see Fig. 2(a2)]. When the feedback strength is greater than a certain value, the microwave linewidth no longer decreases. For example, in Fig. 5(d) with k_f = 1.5 ns^-1, the spin-VCSEL enters into the chaotic oscillation. The power spectrum is broadened significantly and includes a repeating pattern in every 1/τ as a feature of feedback-induced chaos. These results indicate that the feedback strength should be limited to a certain range where the spin-VCSEL operates in P1 oscillations and in such a region, the increase of the feedback strength is beneficial to the decrease of the microwave linewidth. Furthermore, from the power spectrum of Figs. 5(a)-(c), we also find that the second-harmonic ratio (SHR, which is described as the power ratio of the power at the second-harmonic frequency to the power at the fundamental frequency) is gradually increased with the enhancement of the feedback strength, which may degrade the performance of microwave signals due to the increase of phase noise.

To gain more details on the effects of the feedback parameters on the photonic microwave signal, we first display the evolution of the microwave linewidth as a function of the feedback strength k_f for several values of the birefringence split, as shown in Fig. 6(a). As we expect, when the feedback strength is enhanced, the microwave linewidth is significantly reduced for all cases. Until the feedback strength is larger than a certain value, the microwave linewidth increases dramatically. This is mainly because the spin-VCSEL starts to shift out of the P1 dynamics to the complex dynamic regimes. Likewise, the phase noise evolution is calculated in Fig. 6(b). Herein, the phase noise variance is evaluated through integrating the averaged single sideband of the RF spectrum centered at the fundamental frequency over an offset from 3 MHz to 1 GHz and normalized to microwave power [42]. From Fig. 6(b), one can see that, in the microwave linewidth narrowing region, the phase variance first reduces and then slightly oscillates in a certain value with the enhancement of the feedback strength. Out of this region, the phase variance enlarges obviously. These results indicate that, in P1 oscillation regions, the central linewidth of microwave signals can be improved by increasing the feedback strength. Yet, the strong feedback strength does not consistently reduce phase noise due to the appearance of side peaks.

 

Fig. 6. (a) Linewidth and (b) phase noise as a function of the feedback strength, where the delay time is fixed at τ = 5 ns, the intrinsic parameters are set at (γ_s, γ_p) = (35 ns^-1, 30π ns^-1) (black triangle), (35 ns^-1, 44.5π ns^-1) (blue rhombus), (35 ns^-1, 98π ns^-1) (red dots). (c) Linewidth and (d) phase noise as a function of the feedback delay time, where the feedback strength k_f = 0.5 ns^-1 (corresponding to the case of Δf = 31.77 GHz), 0.5 ns^-1 (corresponding to the case of Δf = 45 GHz), and 0.1 ns^-1 (corresponding to the case of Δf = 98 GHz).

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In Figs. 6(c) and 6(d), we also consider the influence of the feedback delay time on the quality of microwave signals. Considering the feedback sensibility in a spin-VCSEL with high birefringence split, which causes the laser to break the P1 oscillation and enter into complex dynamics, we thus set different feedback strengths for three cases of Δf. Here, the feedback strengths are set at k_f = 0.5 ns^-1, 0.5 ns^-1, and 0.1 ns^-1 for the case of Δf = 31.77 GHz, 45 GHz, and 98 GHz, respectively. In the case of Δf = 98 GHz, the complex dynamics are discarded leading to an inconsecutive curve in Figs. 6(c) and 6(d). As we can see from Fig. 6(c), the microwave linewidth shows a continual narrowing with the increase in the feedback delay time for all cases. Therefore, a longer external cavity is beneficial to linewidth narrowing. Nevertheless, a different trend can be observed in the phase noise evolution, as shown in Fig. 6(d). With the increase in the feedback delay time, the phase noise first displays a decrease and then an increase in oscillation. This can be attributed to the increase in the density of the side peaks with expanding the feedback delay time. From the inset of Fig. 6(d), we can find that the SHR significantly drops with increasing the feedback delay time due to the appearance of strong side peaks. Similar phenomena have been found in semiconductor lasers subject to optical injection and optical feedback [42,43].

3.3 P1 with dual-loop feedback

The previous results show that a long delay time is desired for the microwave central linewidth reduction, yet the density of the side peaks grows in magnitude as the delay time increases. To further minimize the microwave central linewidth while remaining the lower phase noise, we introduce the dual-loop feedback scheme to the spin-VCSEL. The two loops are set with little different delay times and thus their side peaks are misaligned, triggering the Vernier effect [42]. The total feedback strength is k_f = k_f1 + k_f2 and the feedback delay time difference is τ₂₁= τ₂ - τ₁. In Fig. 7, the feedback delay times are expanded to τ₁ = 17 ns and τ₂ = 20 ns. Beginning with Fig. 7(a), the loop of τ₁ is connected while the loop of τ₂ is broken, where k_f1 = 0.7 ns^-1 and k_f2 = 0. The optical feedback with the long delay time narrows the microwave linewidth to Δf₀ = 20 kHz. Likewise, in Fig. 7(b), the loop of τ₂ is turned on while the loop of τ₁ is disconnected, where k_f1 = 0 and k_f2 = 0.7 ns^-1. The microwave linewidth reduces to Δf₀ = 18 kHz. The long feedback delay time in Figs. 7(a) and 7(b) lead to the much narrower central linewidth, in comparison with the central linewidth of 75 kHz in Fig. 5(b) for a short delay time. Nonetheless, the use of long delay times in Figs. 7(a) and 7(b) cause the dense side peaks separated by 1/τ₁ and 1/τ₂, respectively. Consequently, the feedback with a long delay time allows a smaller central linewidth compared with the scenario of a short delay time, but it will result in strong side peaks that worsen the phase variance. To solve the problem, both loops are switched on [see Fig. 1(a)] and we can clearly observe that the side peaks are suppressed in Fig. 7(c). This can be attributed to the fact that the two delay times with slight differences trigger the Vernier effect [44,45], which promotes the reduction of the phase variance. In the case of dual-loop feedback, the feedback strength is set equally, e.g., in Fig. 7(c), k_f1 = k_f2 = 0.35 ns^-1, and the total feedback strength k_f = 0.7 ns^-1. From Fig. 7(c), the central linewidth is reduced to 10 kHz, and the phase noise variance drops to the order of 0.001 rad². We further increase both feedback strengths to k_f1 = k_f2 = 0.5 ns^-1, Fig. 7(d) exhibits a further decrease of the microwave linewidth down to 4 kHz and the phase variance is less than 0.001 rad². Therefore, dual-loop feedback outperforms single-loop feedback due to its advantages in lowering the phase variance or minimizing the microwave linewidth.

 

Fig. 7. Microwave power spectra of the spin-VCSEL with dual-loop feedback. The feedback strength (k_f1, k_f2) = (a) (0.7 ns^-1, 0), (b) (0, 0.7 ns^-1), (c) (0.35 ns^-1, 0.35 ns^-1), and (d) (0.5 ns^-1, 0.5 ns^-1). The feedback delay time is fixed at (τ₁, τ₂) = (17 ns, 20 ns). The intrinsic parameters are fixed at (γ_s, γ_p) = (35 ns^-1, 30π ns^-1).

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To quantify the suppression of side peaks, we calculate the phase variance with the variation of k_f, as shown in Fig. 8, where the feedback delay times are fixed at (τ₁, τ₂) = (17 ns, 20 ns). For the scenario of single feedback, only one loop is connected, e.g., (i) k_f1 = k_f and k_f2 = 0 (black rhombus); (ii) k_f1 = 0 and k_f2 = k_f (gray triangle). From Fig. 8, one can observe that, regardless of which loop is turned on, the phase variance can be constantly reduced with the increase in the feedback strength k_f, and finally rises when the laser enters into the complex dynamics. For the scenario of dual-loop feedback, both loops are connected and set with equal strengths, i.e., k_f1 = k_f2 =k_f/2. Compared with the case of the single loop, the dual-loop feedback shows a smaller phase noise variance due to the suppression of side peaks. Moreover, we also can find that the spin-VCSEL operates in P1 oscillations over a larger range of k_f for the dual-loop feedback. Before the P1 is destabilized, the phase variance can be reduced to 0.001 rad², which is about one order of magnitude smaller than that of the single loop feedback. Although the dual-loop does not fully eliminate all side peaks [see Figs. 7(c) and 7(d)], it can offer a smaller phase noise variance compared to that from the scenario of single feedback and expand the scope of the total feedback strength for P1 dynamics.

 

Fig. 8. Phase variance as a function of total feedback strength k_f. For the case of dual-loop feedback, the feedback strengths are set as (k_f1, k_f2) = (k_f/2, k_f/2) and the feedback delay times are set as (τ₁, τ₂) = (17 ns, 20 ns).

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The use of dual-loop feedback inevitably leads to a complex parameter set, which significantly affects the performance of microwave signals. Therefore, we investigate the influence of the feedback strength ratio between the two feedback loops on the phase variance in Fig. 9(a), where the feedback delay times are fixed at (τ₁, τ₂) = (17 ns, 20 ns) and total feedback strengths are 0.2 ns^-1, 0.4 ns^-1, and 0.6 ns^-1. From Fig. 9(a), one can see that the minimization of the phase variance can be achieved at the ratio k_f1/k_f2 of ∼1. This can be concluded as the side peaks suppression because of the competition of the external-cavity modes. Additionally, when the ratio k_f1/k_f2 deviates from 1, the phase variance is progressively increased due to the fact that the dual-loop feedback evolves gradually similar to single feedback which degrades the suppression of side peaks. These results indicate that the feedback strength should be evenly distributed between the two feedback loops to obtain the minimum P1 phase variance.

 

Fig. 9. (a) Phase variance as a function of the feedback strength ratio k_f1/k_f2, where the feedback delay time (τ₁, τ₂) = (17 ns, 20 ns). (b) Phase variance as a function of the feedback delay time ratio τ₁/τ₂, where the feedback delay time τ₁ is fixed at 30 ns and τ₂ is varied. The total feedback strengths are fixed at 0.2 ns^-1, 0.4 ns^-1, and 0.6 ns^-1 as the label.

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Likewise, we also consider the influence of changing the feedback delay time of two feedback loops, as shown in Fig. 9(b). Here, the feedback delay time τ₁ is set at 30 ns, and τ₂ is varied. The feedback strengths of two loops are set equal, i.e., k_f1 = k_f2, and total feedback strengths are 0.2 ns^-1, 0.4 ns^-1, and 0.6 ns^-1. From Fig. 9(b), we can see a relatively large phase variance occurring in some special ratios, e.g., τ₂/τ₁ = 1/3, 1/2, and 2/3, which leads to strong side peaks at the common multiples of the reciprocals of the delays. Such phenomena were also observed in the DFB and conventional VCSEL subject to optical injection and dual-loop optical feedback systems [42,43]. Moreover, when the delay time ratio is close to 1, the phase variance sharply increases for all three cases of total feedback strengths. This is mainly because the dual-loop feedback is reduced to single-loop feedback with strong side peaks. Overall, dual-loop feedback can suppress the side peak by leveraging the Vernier effect, which supports deploying a long delay time to narrow the microwave linewidth and keeping a small phase variance. However, we should point out that the feedback strength of the two loops is preferentially set to be equal and the feedback delay is incommensurate.

4. Conclusion

In conclusion, we have systematically investigated the nonlinear P1 dynamics of the optically pumped spin-VCSEL subject to optical feedback. For a solitary spin-VCSEL, the P1 oscillation frequency is widely adjustable beyond the RO frequency, which strongly depends on the birefringence γ_p. More importantly, we have demonstrated that the f₀ can be increased (decreased) by adding a positive (negative) axial magnetic field when the birefringence γ_p is fixed. Combining two experimental controllable parameters (i.e., η, P), the tunability of the P1 microwave frequency is significantly enhanced. However, the introduction of an axial magnetic field degenerates the microwave linewidth in the edge of the Hopf bifurcation. More importantly, optical feedback is further employed to boost the performance of the P1 microwave signals. The results show that, for the scenario of single-loop optical feedback, the microwave linewidth can be sharply decreased by increasing the feedback strength and/or delay time, while the phase noise oscillation upraises with the increase of the feedback delay time. By adding dual-loop feedback, the incommensurate delay time ratio availably suppresses the side peaks in the power spectra by the vernier effect, which allows the P1 linewidth minimization at a long time and holds a small phase noise.