1. Introduction

Microwave photonics offers original solutions to overcome imminent bottlenecks in electronic circuits for the generation of microwaves on an optical carrier, the transmission of information, and the sensing of the environment [1]. Photonic microwaves have found applications in radio-over-fiber (RoF), THz spectroscopy, photonic based remote sensing, beamforming, radio astronomy, and all-optical information and signal processing [2–6].

Ideal photonic microwave sources have high microwave frequencies, wide microwave frequency-tunability, narrow microwave linewidth or low phase noise, and simple or integrated circuitry [7,8]. Direct and external modulation of a semiconductor laser generates photonic microwaves with linewidths equal to the linewidth of the electrical signal that is used to drive the modulator. Such methods utilize high-frequency electronics and have a limited microwave frequency range due to the relaxation resonance of the semiconductor laser or due to the frequency response of the modulator [9]. On the cost of increasing the system complexity, optoelectronic oscillators (OEOs) overcome this microwave frequency limitation and avoid the electrical local oscillator by generating the photonic microwave through one of the resonant mode frequencies of the optoelectronic oscillation loop [10]. To entirely omit the electronics, optical heterodyne of two lasers with different wavelengths can be used to generate high-frequency photonic microwaves with wide tunable capabilities. However, due to the uncorrelated phases of the two optical beams, the generated photonic microwave has a large microwave linewidth. Adding a phase-locked loop stabilizes the photonic microwave oscillation generated by optical heterodyne but evidently increases the complexity of the system [11]. A mode-locked laser with the two modes separated at the desired microwave frequency generates without external electronics a photonic microwave with a narrow linewidth. However, this approach limits the tunability of the microwave frequency because the mode separation of a mode-locked laser is determined by the fixed optical length of the laser cavity [12].

Optically injected semiconductor lasers generate self-sustained oscillations in the microwave region known as period-one (P1) dynamics. The nonlinear dynamics of the semiconductor laser are invoked by perturbing the laser at critical operating points where the relaxation resonance of the semiconductor laser becomes undamped as it crosses a Hopf bifurcation. The stable injection-locked laser generates limit cycles of P1 oscillations through a Hopf bifurcation and reaches chaos through a cascaded period-doubling bifurcation. Therefore, these microwave frequencies are inherently distinct from the microwave oscillations generated by optical heterodyne and are characteristically different [13]. The P1 oscillation generated by an optically injected semiconductor laser is widely tunable and can reach frequencies up to ten times the relaxation resonance frequency of the un-injected solitary semiconductor laser. Furthermore, the optical injection setup is fairly simple, and the generated microwave frequency can be simply tuned by varying the detuning frequency, injection strength, or bias current of the master and/or slave lasers [14]. Spontaneous emission noise of the two lasers and time-dependent fluctuations in the operating parameters cause the P1 oscillation to have a broad microwave linewidths typically on the megahertz order of magnitude [15–17]. Extensive research efforts have been conducted to stabilize the P1 oscillation induced by optical injection. An external microwave source that is tuned to the P1 oscillation frequency or its subharmonics double-locks and stabilizes the P1 oscillation [18–20]. The phase of the microwave subcarrier can be phase-locked to an external microwave by adding a phase-locked loop in the optical injection setup to further stabilize the P1 oscillation [21]. To get rid of the external microwave reference, self-locking methods, including optoelectronic feedback [22,23], optical feedback [24,25], dual-loop optical feedback [26], and polarization-rotated optical feedback [27], have been demonstrated to lock the P1 oscillation and narrow its microwave linewidth by two to three orders of magnitude.

Recently, operating points of the optically injected laser that show low sensitivity to the time-dependent fluctuations in the operating parameters have been demonstrated [28]. At specific operating conditions, the nonlinear effects are exploited to mitigate the perturbation-induced noise in the system [29]. These low-sensitivity operating points are identified at regions where the P1 oscillation undergoes a frequency minimum with respect to the corresponding operating parameter. Three distinct types of low-sensitivity operating points are identified: low sensitivity to injection-strength fluctuations, low sensitivity to detuning-frequency fluctuations, and low sensitivity to bias-current fluctuations. The laser cavity configuration, the intrinsic properties of the semiconductor medium, and the noise intensity and frequency all play certain roles in the characteristics of the generated low-sensitivity operating points [30,31].

In this work, the effects of the intrinsic laser noise sources of the optical injection system on the characteristics of the microwave linewidth of the P1 oscillation are investigated. Unlike previous efforts of related works [16,32], the field noise of both the slave and the master lasers are considered and systematically investigated in the present study. Moreover, the effects of the operating conditions of the optical injection system on the microwave linewidth of the P1 oscillation when each noise term is added are analyzed. Therefore, this work complements ongoing efforts to stabilize the P1 oscillations induced by optically injected semiconductor lasers.

The paper is organized as follows. After this introductory section, Section 2 describes the theoretical model used to conduct the numerical experiments. The effects of the different noise sources and the effect of the noise strength on the microwave linewidth of the P1 oscillation are analyzed in Sections 3 and 4, respectively. Mapping of the microwave linewidth of the P1 oscillation as a function of the operational parameters for the different noise sources is given and discussed in Section 5. Finally, a conclusion summarizing the major results of this study is drawn in Section 6.

2. Theoretical model

The nonlinear dynamics of an optically injected semiconductor laser can be adequately described by a set of coupled nonlinear equations relating the intracavity optical field to the carrier density. In the following equations, the superscripts M and S represent the master laser and slave laser, respectively. To account for the master laser noise, the master laser is modeled as a free-running semiconductor laser with field noise. The carrier noise is not shown in the model because it has a very narrow bandwidth compared to the optical noise contributed by the spontaneous emission; when needed, it can be considered by adding a noise term to the carrier equation of a laser, as discussed in Section 3. The master laser is described by a set of three coupled nonlinear equations, where a_r^M and a_i^M respectively represent the real and imaginary parts of the normalized complex field amplitude of the master laser, and ñ^M is the normalized carrier density of the master laser [33]. The terms ${F}_\textrm{r}^{\textrm{M}}$ and ${F}_\textrm{i}^{\textrm{M}}$ are the real and imaginary parts, respectively, of the field noise of the master laser.

Here, for simplicity, the intrinsic parameters of the master and slave lasers are taken to be the same, which have a linewidth enhancement factor of b^S = b^M = 3.2, a gain saturation factor of b′^S = b′^M = 3.2, a cavity decay rate of γ_c^S= γ_c^M = 5.36 × 10¹¹ s⁻¹, a spontaneous carrier relaxation rate of γ_s^S= γ_s^M = 5.96 ×10⁹ s⁻¹, a differential carrier relaxation rate of γ_n^M = $\tilde{J}^{M}$ · 6.162 ×10⁹ s⁻¹ for the master laser and γ_n^S = $\tilde{J}^{S}$· 6.162 ×10⁹ s⁻¹ for the slave laser, and a nonlinear carrier relaxation rate of γ_p^M = $\tilde{J}^{M}$ · 1.536 ×10¹⁰ s⁻¹ for the master laser and γ_p^S = $\tilde{J}^{S}$ · 1.536 ×10¹⁰ s⁻¹ for the slave laser [35]. The intrinsic parameters adopted here are all experimentally extracted from an InGaAsP/InP DFB semiconductor laser using a well-established four-wave mixing technique [36]. The relaxation resonance frequencies of the two semiconductor lasers with the aforementioned intrinsic parameters are 10.25 GHz for a normalized bias current of $\tilde{J}^{S}$ = $\tilde{J}^{M}$ = 1.222.

To account for the intrinsic laser noise of the master and slave lasers due to spontaneous emission, complex Langevin fluctuating force terms are included in Eqs. (1)–(6). The real and imaginary parts of the complex field noise are uncorrelated and the Langevin force at different times is also uncorrelated. Therefore, for a laser operating above the laser threshold, the normalized noise parameters are related to the linewidth of the laser through the following averages [16,33,37–39]:

Here, F_r and F_i are respectively the normalized real and imaginary Langevin noise-source terms due to the field noise of the laser. The full width at half-maximum (FWHM) laser linewidth, Δv, represents the free-running slave laser optical linewidth or the free-running master laser optical linewidth by Δv_S or Δv_M, respectively. In this work, the FWHM optical linewidth of the free-running slave laser and that of the free-running master laser are varied in the range of Δν = 1–200 MHz to approach the optical and microwave linewidths that were experimentally measured for different lasers [22,40,41].

For the numerical experiments, a second-order Runge-Kutta integration on Eqs. (1)–(6) is used with a duration of 4 µs and an integration time step of 0.95 ps, resulting in a spectral resolution of 250 kHz. The spectral resolution of the numerical simulation is an order of magnitude smaller than the narrowest observed optical and microwave linewidths and therefore is sufficient for the current purpose. The optical and power spectra of the slave laser output are calculated by taking the Fourier transform of the resulting time series. The optical and microwave linewidths are then respectively extracted from the generated optical and power spectra using a Lorentzian fitting algorithm.

The differential and nonlinear relaxation rates along with the optical linewidths of the slave and master lasers vary with the normalized bias currents of the respective lasers; therefore, as the values of $\tilde{J}^{S}$ and $\tilde{J}^{M}$ are varied, this dependency on the normalized bias current has to be taken into account in the theoretical model. Figure 1 shows the relaxation resonance frequency and the free-running optical linewidth of an un-injected semiconductor laser as a function of the normalized bias current. The aforementioned relaxation rates have a linear relationship with the normalized bias current, thus resulting in the relaxation resonance frequency shown in Fig. 1 [36]. The optical linewidth of the laser is inversely proportional to the normalized bias current, as illustrated in Fig. 1 [40]. In the following discussion, the normalized bias current of the master laser is kept constant at $\tilde{J}^{M}$ = 1.222, while that of the slave laser is varied as needed.

 

Fig. 1. Relaxation resonance frequency and the optical linewidth of the free-running semiconductor laser as functions of the normalized bias current.

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3. Influence of the laser noise sources on the optical and microwave spectral linewidths

To demonstrate the effects of the intrinsic laser noise of the master and slave lasers on the generated P1 oscillation frequency, the optically injected laser is first driven into a P1 dynamic to generate a microwave frequency, f₀. The slave laser has a free-running optical frequency of v₀, but it is injection-locked by the master laser at the optical frequency of v_i at a detuning frequency of f = v_i – v₀. The optical spectrum of the slave laser is plotted in the left column of Fig. 2, where the optical frequency is offset with respect to the free-running frequency v₀ of the slave laser. In a P1 dynamic, the slave laser is locked at the frequency of the master laser and the injection signal at this frequency is regeneratively amplified with the characteristics of a frequency offset f of the detuning frequency and a linewidth Δv_i of the master laser. The principal oscillation occurs at an optical frequency of v_i − f, which has a frequency offset of f − f₀ and a linewidth of Δv₀. In the microwave spectrum shown in the middle column of Fig. 2, the P1 microwave frequency at f₀ has a linewidth of Δf₀. In the carrier spectrum shown in the right column of Fig. 2, the P1 microwave frequency at f₀ has a linewidth of Δf_n. These linewidths are calculated and compared for different operating conditions.

 

Fig. 2. Optical spectra (left column), power spectra (middle column), and carrier spectra (right column) of the optically injected semiconductor laser at (ξ, f) = (0.18, 16 GHz) exciting a P1 dynamic at f₀ = 25 GHz. The frequency of the optical spectrum is offset with respect to the free-running optical frequency of the slave laser. The slave laser is injection-locked at the frequency v_i of the master laser. The principal oscillation occurs at an optical frequency of v_i − f₀, which has a frequency offset of f − f₀ as seen in the left column of this figure. (a) No laser noise is considered; (b) only the slave laser noise is considered, with Δv_S = 50 MHz; (c) only the master laser noise is considered, with Δv_M = 50 MHz; (d) only the carrier noise of the slave laser is considered; (e) both slave laser noise, with Δv_S = 50 MHz, and master laser noise, with Δv_M = 50 MHz, are considered.

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Figure 2 shows the influence of the different noise sources on the optical, microwave, and carrier frequency components when the slave laser is subject to optical injection of (ξ, f) = (0.18, 16 GHz) generating a microwave frequency at f₀ = 25 GHz. The horizontal axis of the optical spectrum (left column) represents the frequency offset from the free-running optical frequency of the slave laser. The horizontal axis of the power spectrum (middle column) and carrier spectrum (right column) is centered at the generated P1 frequency of f₀ = 25 GHz.

Figure 2(a) shows the clean optical and power spectra at this injection condition when the noise of both the master and the slave lasers is suppressed. This optical injection excites a P1 dynamic showing multiple optical frequency components seen in Fig. 2(a-i), which are equally spaced at an offset frequency of 25 GHz, corresponding to the microwave frequency, f₀, shown in the power spectrum seen in Fig. 2(a-ii).The principal oscillation frequency component at a frequency offset of f − f₀ and the regeneratively amplified component at a frequency offset of f are at least 10 dB higher than the other frequency components of the optical spectrum. The resulting 3-dB linewidths of the optical and microwave frequency components are 250 kHz, which is the limit of our numerical spectral resolution and theoretically represents a zero linewidth. Therefore, we show that Δv_i = 0 Hz, Δv₀ = 0 Hz, and Δf₀ = Δv₀ = 0 Hz in Figs. 2(a-i) and 2(a-ii) when noise from both lasers are suppressed. Figure 2(b) shows the optical and power spectra when the field noise of the slave laser is considered while that of the master laser is suppressed. The optical linewidth of the free-running slave laser is set at Δv_S = 50 MHz while that of the free-running master laser is kept at Δv_M = 0 Hz. A broad noise pedestal appears, and broadening around each spectral component is observed except at the regeneratively amplified component, which is still narrower than our frequency resolution limit when Δv_i = Δv_M = 0 Hz. The 3-dB linewidth of the principal oscillation frequency component broadens to Δv₀ = 13.5 MHz, which is narrower than the free-running slave laser linewidth set at Δv_s = 50 MHz, as shown in Fig. 2(b-i). The microwave linewidth of the P1-frequency also broadens to Δf₀ = Δv₀ = 13.5 MHz corresponding to the linewidth of the principal oscillation frequency component, as shown in Fig. 2(b-ii). Figure 2(c) shows the optical and power spectra when the field noise of the master laser is considered while that of the slave laser is suppressed. The optical linewidth of the free-running slave laser is set at Δv_S = 0 Hz while the optical linewidth of the free-running master laser is set at Δv_M = 50 MHz. Broadening around each spectral component is observed, with the linewidth of the regeneratively amplified component broadening to Δv_i = 50 MHz, which is equal to the optical linewidth of the free-running master laser. The phase noise of the regeneratively amplified component couples to the component at the principal oscillation frequency, which broadens to Δv₀ = 25 MHz. The microwave linewidth of the P1 oscillation broadens to Δv₀ = 15 MHz, which is narrower than the free-running master laser optical linewidth set at Δv_M = 50 MHz and also narrower than the linewidth of every optical frequency component. The narrow microwave linewidth of the P1 oscillation indicates that the optical frequency components of the P1 oscillation are phase correlated. To account for amplified spontaneous emission in the weak side modes that induce carrier fluctuations, a Langevin noise term is added to the carrier equation, Eq. (6), of the slave laser [33]. Figure 2(d) shows the optical and power spectra when the carrier noise of the slave laser is considered while the field noise of the master and slave lasers is suppressed. Although a small noise pedestal appears due to carrier noise, no broadening around the spectral components is observed; therefore, the contribution of the carrier noise is neglected throughout this work. Figure 2(e) shows the optical and power spectra when the field noise of both the master and the slave lasers is considered. The optical linewidth of the free-running slave laser is set at Δv_S = 50 MHz and that of the free-running master laser is set at Δv_M = 50 MHz. Broadening around each spectral component is observed, with the linewidth of the regeneratively amplified optical component extending to Δv_i = 50 MHz, equal to the optical linewidth of the free-running master laser. The optical component at the principal oscillation frequency broadens to Δv₀ = 43.5 MHz. The microwave linewidth of the P1 oscillation broadens to Δv₀ = 28 MHz, which is narrower than optical linewidths of the free-running lasers, Δv_M = 50 MHz and Δv_S = 50 MHz. The carrier spectra show the influence of the different noise sources on the carrier linewidths. Linewidth characteristics that are similar to those of the microwave spectra are observed in the linewidths of the carrier spectra. The narrowing of the microwave linewidth of the P1 oscillation below the optical linewidth of the free-running slave laser and that of the free-running master laser is attributed to nonlinear frequency mixing of the various frequency components of the optical spectrum.The influence of the field noise of the master laser on the microwave linewidth of the P1 oscillation is significant and cannot be neglected, as demonstrated in Fig. 2. The linewidth of the regeneratively amplified optical component follows the optical linewidth of the free-running master laser. The linewidth narrowing effect depends not only on the noise sources but also on the strengths of these sources and the operational parameters of the optical injection system.

4. Influence of the lasers noise strength on the optical and microwave spectral linewidths

The linewidth of the typical single-mode semiconductor laser ranges from 1 MHz to 200 MHz depending on the spontaneous emission noise of the laser, the type of the laser, the intracavity laser power, the losses of the optical resonator, and the cavity round-trip time [9]. Due to nonlinear frequency mixing between the various optical frequency components, a semiconductor laser under optical injection can generate microwave frequencies that are narrower than the free-running linewidth of the laser, as demonstrated in Fig. 2. Figure 3 demonstrates the effects of the noise strengths of the master laser and the slave laser on the optical linewidth of the principal oscillation frequency component (red diamonds), Δv₀, the optical linewidth of the regeneratively amplified frequency component (blue squares), Δv_i, and the microwave linewidth of the P1-frequency (black circles), Δf₀, by showing these linewidths as a function of the free-running master and slave laser linewidths.

 

Fig. 3. Optical and microwave linewidths of the optically injected semiconductor laser at (ξ, f) = (0.18, 16 GHz) exciting a P1 dynamic at f₀ = 25 GHz (a) as a function of the free-running master laser linewidth, Δv_M, and (b) as a function of the free-running slave laser linewidth, Δv_S.

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Figure 3(a) shows the linewidths Δv_i, Δv₀, and Δf₀ when the linewidth of the free-running slave laser is fixed at Δv_S = 50 MHz while the free-running master laser linewidth Δv_M is varied from 10 MHz to 200 MHz. The linewidth Δv_i of the regeneratively amplified optical component is equal to the free-running master laser linewidth, Δv_M, as it is varied from 10 MHz to 200 MHz, as illustrated in blue squares in Fig. 3(a). The 3-dB optical linewidth Δv₀ of the principal oscillation frequency component broadens as Δv_M increases but is narrower than the free-running master laser linewidth. Similarly, the microwave linewidth Δf₀ of the P1 oscillation broadens as Δv_M increases and is narrower than the linewidths of the optical frequency components. Figure 3(b) shows the linewidths Δv_i, Δv₀, and Δf₀ when the free-running master laser linewidth is fixed at 50 MHz while the free-running slave laser linewidth is varied from 10 MHz to 200 MHz. The linewidth of the regeneratively amplified optical component, Δv_i, remains constant and is equal to the free-running master laser linewidth, Δv_M = 50 MHz, as the free-running slave laser linewidth is varied from 10 MHz to 200 MHz, as illustrated by the blue squares of Fig. 3(b). The linewidth of the component at the principal oscillation frequency and the microwave linewidth of the P1 oscillation both broaden as Δv_S is increased. The microwave linewidth of the P1 oscillation is narrower than the linewidth of the optical component at the principal oscillation frequency for small Δv_S values, and the two linewidths approach each other for large Δv_S values, as shown in Fig. 3(b). This demonstrates that the linewidth of the regeneratively amplified optical component, Δv_i, is affected only by the free-running master laser linewidth regardless of the free-running slave laser linewidth. On the other hand, the linewidth Δv₀ of the optical component at the principal oscillation frequency and the microwave linewidth Δf₀ of the P1 oscillation are affected by both the free-running master laser linewidth and the free-running slave laser linewidth. Due to the phase correlation among the optical components, the microwave linewidth of the P1-frequency is always narrower than the optical linewidth of the principal oscillation component.

5. Influence of the operational parameters on the optical and microwave spectral linewidths

The nonlinear dynamics generated by an optically injected semiconductor laser depend on the operating conditions of the optical injection system and on the intrinsic semiconductor laser parameters [42]. Similarly, the power and frequency characteristics of the P1 dynamic excited by optical injection depends on the operating conditions of the optical injection system and on the intrinsic semiconductor laser parameters [30,34]. To demonstrate the linewidth characteristics of the P1 dynamics, mapping of the microwave linewidth of the P1 oscillation as a function of the operational parameters is conducted.

Figure 4 shows the mapping of the P1 oscillation frequency and its 3-dB microwave linewidth for different noise sources as a function of the injection strength and the detuning frequency while the bias current of the slave laser is fixed at $\tilde{J}^{S}$ = 1.222. At certain operating conditions, the semiconductor laser under optical injection generates chaotic oscillations or nonoscillatory stable injection-locked fixed-point states (uncolored), which are separated from the P1 dynamics (colored) by dense curves for various types of bifurcations. The P1 frequency is represented by thick contour curves labeled in GHz, whereas the colored regions show the microwave linewidth of the P1 oscillation in MHz, with the yellow curve indicating a microwave linewidth equal to 20 MHz. The red dots show the operating points of maximum P1-frequency power for a specified injection strength. Figure 4(a) shows the microwave linewidth of the P1 oscillation when the field noise of the master laser is considered while that of the slave laser is suppressed. The free-running optical linewidth of the slave laser is set at Δv_S = 0 Hz, and the free-running optical linewidth of the master laser is set at Δv_M = 50 MHz. The operating points of the minimum linewidth are situated at the highly nonlinear region close to the locations where the P1 oscillation undergoes a frequency minimum with respect to the detuning frequency and the P1-frequency power is at its maximum. The narrowest microwave linewidth reaches 2.25 MHz at the operating point of (ξ, f) = (0.12, 5 GHz), in the region of relatively low injection strength and small frequency detuning. The microwave linewidth of the P1 oscillation broadens as the injection strength or the detuning frequency is increased to a P1 frequency beyond the relaxation resonance frequency of the slave laser, reaching the optical linewidth of the free-running master laser set at Δv_M = 50 MHz, as illustrated in Fig. 4(a). Figure 4(b) shows the microwave linewidth of the P1 oscillation when the field noise of the slave laser is considered while that of the master laser is suppressed. The free-running optical linewidth of the slave laser is set at Δv_S = 50 MHz, and the free-running optical linewidth of the master laser is set at Δv_M = 0 Hz. The narrowest microwave linewidth reaches 7 MHz at the operating point of (ξ, f) = (0.12, 6 GHz), which is broader than the narrowest microwave linewidth when only the master laser field noise is considered. Similar trends are observed, as compared to Fig. 4(a), in the microwave linewidth of the P1 oscillation as the injection strength or the detuning frequency is increased to generate a P1 frequency beyond the relaxation resonance frequency of the slave laser, as illustrated in Fig. 4(b). Figure 4(c) shows the microwave linewidth of the P1 oscillation when the field noise of both the slave and the master lasers is considered by setting the free-running optical linewidths of the slave laser and the master laser at Δv_S = 50 MHz and Δv_M = 50 MHz, respectively. The narrowest microwave linewidth reaches 10 MHz at the operating point of (ξ, f) = (0.12, 5 GHz), which is higher than the narrowest microwave linewidth when only the field noise of the master laser or only the field noise of the slave laser is considered. Similar trends are observed, as compared to Figs. 4(a) and (b), in the microwave linewidth of the P1 oscillation as the injection strength or the detuning frequency is increased to generate a P1 frequency beyond the relaxation resonance frequency of the slave laser, as illustrated in Fig. 4(c). The narrow microwave linewidth of the P1 oscillation below the optical linewidths of the slave and master lasers at low injection strengths and small values of frequency detuning is due to the phase correlation among the optical frequency components. As the injection strength or the detuning frequency is increased, the phases of the optical frequency components become uncorrelated and broadening is observed in the microwave linewidth of the P1 oscillation. The phase decorrelation among the optical frequency components is attributed to the reduced influence of the carriers on the nonlinear dynamics as the P1 frequency is increased, which signifies reduced coupling between the refractive index of the gain medium and the optical gain [27]. It has been demonstrated numerically [16] and experimentally [17] that the operating point that generates a P1 oscillation of a minimal microwave linewidth due to field noise is located very close to the point where the P1 microwave power is maximized. It has also been demonstrated numerically [29] and experimentally [43] that the operating point that generates a P1 oscillation of low sensitivity to time-dependent fluctuations in the operating parameters is located very close to the point where the P1 oscillation undergoes a frequency minimum with respect to the corresponding operating parameter. Moreover, from Fig. 4, at low injection strengths and small values of detuning frequency, the operating point of the maximum P1 microwave power and the operating point where the P1 oscillation undergoes a frequency minimum with respect to the detuning frequency are close to each other. As the P1 frequency is increased, the operating point of the minimum P1 frequency with respect to the detuning frequency and the operating point of the maximum P1-frequency power diverge. This is illustrated in the contour curves and the red dots of Fig. 4.

 

Fig. 4. Mapping of the 3-dB microwave linewidth Δf₀ of the P1 oscillation induced by an optically injected semiconductor laser as a function of the injection strength and the detuning frequency while the bias current is fixed at $\tilde{J}^{S}$ = 1.222. (a) Only the master laser noise is considered, with Δv_M = 50 MHz and Δv_S = 0 Hz; (b) only the slave laser noise is considered, with Δv_S = 50 MHz and Δv_M = 0 Hz; (c) both slave laser noise and master laser noise are considered, with Δv_S = 50 MHz and Δv_M = 50 MHz. The P1 frequency is represented by thick contour curves in GHz, whereas the colored regions show the microwave linewidth of the P1 oscillation in MHz with the yellow curves indicating a 3-dB microwave linewidth equal to 20 MHz. Uncolored regions represent complex dynamics and stably injection-locked fixed-point states, which are separated from the P1 dynamics by dense curves for the period-doubling bifurcation line and the Hopf bifurcation line, respectively. The red dotted curve shows the maximum P1-frequency power for a specified injection strength.

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Figure 5 shows mapping of the P1 oscillation frequency and its 3-dB microwave linewidth for different noise sources as a function of the injection strength and the normalized bias current of the slave laser while the detuning frequency is fixed at f = 10 GHz. The conventions used in Fig. 4 to represent the nonlinear dynamics and the microwave linewidth are used in Fig. 5. The free-running optical linewidth of the slave laser is adjusted as the slave laser bias current is varied, showing an inversely proportional relationship generating an optical linewidth of Δv_S = 50 MHz when $\tilde{J}^{S}$ = 1.222, as demonstrated in Fig. 1. The linearly proportional relationship between the relaxation rates and the normalized bias current is also considered in the numerical simulation according to γ_n^S= $\tilde{J}^{S}$ · 6.162 ×10⁹ s⁻¹ and γ_p^S= $\tilde{J}^{S}$ · 1.536 ×10¹⁰ s⁻¹. The red dotted curve shows the maximum P1-frequency power for a specified injection strength.

 

Fig. 5. Mapping of the 3-dB microwave linewidth Δf₀ of the P1 oscillation induced by an optically injected semiconductor laser as a function of the injection strength and the bias current of the slave laser while the detuning frequency is fixed at f = 10 GHz. (a) Only the master laser noise is considered, with Δv_M = 50 MHz and Δv_S = 0 MHz; (b) only the slave laser noise is considered, with Δv_S = 50 MHz and Δv_M = 0 MHz; (c) both slave laser noise and master laser noise are considered, with Δv_S = 50 MHz and Δv_M = 50 MHz. The P1 frequency is represented by thick contour curves in GHz, whereas the colored regions show the 3-dB microwave linewidth of the P1 oscillation in MHz with the yellow curves indicating a 3-dB microwave linewidth equal to 20 MHz. Uncolored regions represent complex dynamics and stably injection-locked fixed-point states, which are separated from the P1 dynamics by dense curves for the period-doubling bifurcation line and the Hopf bifurcation line, respectively. The red dotted curve shows the maximum P1-frequency power for a specified injection strength.

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Figure 5(a) shows the microwave linewidth of the P1 oscillation when the field noise of the master laser is considered by setting Δv_M = 50 Hz while that of the slave laser is suppressed by setting Δv_S = 0 Hz. The narrowest microwave linewidth of the P1 oscillation reaches 2 MHz at the operating point of (ξ, $\tilde{J}^{S}$) = (0.14, 2.1), which falls on the curve of maximum P1-frequency power shown as the red dotted curve in Fig. 5(a). The microwave linewidth of the P1 oscillation broadens as the injection strength or bias current of the slave laser is varied away from the point where the microwave linewidth is at its minimum. Figure 5(b) shows the microwave linewidth of the P1 oscillation when the field noise of the slave laser is considered by setting Δv_S = 50 MHz while that of the master laser is suppressed with Δv_M = 0 Hz, for a normalized bias current of $\tilde{J}^{S}$ = 1.222. The narrowest microwave linewidth of P1 oscillation reaches 4 MHz at the operating point of (ξ, $\tilde{J}^{S}$) = (0.16, 1.8), which is twice the narrowest linewidth when only the field noise of the master laser is considered. Similar trends are observed, as compared to Fig. 5(a), in the microwave linewidth of the P1 oscillation as the injection strength or the bias current of the slave laser is varied away from the minimum-linewidth point, as illustrated in Fig. 5(b). Note that as the bias current of the slave laser is increased, the free-running optical linewidth of the slave laser becomes narrower, as shown in Fig. 1, yet the minimum-linewidth operating point of the P1 oscillation is still located at the point where the P1-frequency power is maximized. Figure 5(c) shows the microwave linewidth of the P1 oscillation when both the field noise of the slave laser and that of the master laser are considered, with Δv_S = 50 MHz and Δv_M = 50 MHz for a normalized bias current of $\tilde{J}^{S}$ = 1.222. The narrowest microwave linewidth of the P1 oscillation reaches 6 MHz at the operating point of (ξ, $\tilde{J}^{S}$) = (0.14, 2.1), which falls on the curve of the maximum P1-frequency power shown as the red dotted curve in Fig. 5(c). Similar trends are observed, as compared to Figs. 5(a) and 5(b), in the 3-dB microwave linewidth of the P1 oscillation as the injection strength or the bias current of the slave laser is varied away from the narrowest-linewidth operating point, as illustrated in Fig. 5(c). The operating points where the P1 oscillation undergoes a frequency minimum with respect to the injection strength also show a reduction in the microwave linewidth although it is still broader than the linewidth found at the minimum-linewidth operating point. These low-sensitivity operating points only suppress the effect of the fluctuations in the injection strength, whereas fluctuations in the other operating parameters still play a role. The operating point where the P1 oscillation undergoes a frequency minimum with respect to the bias current of the slave laser appears at a location close to the point where the P1-frequency power is at a maximum, as shown in the contour curves and the red dotted line of Fig. 5 [29].

Figure 6 shows mapping of the P1 oscillation frequency and its 3-dB microwave linewidth for different noise sources as a function of the normalized bias current of the slave laser and the detuning frequency when the injection strength is fixed at ξ = 0.15. The conventions used in Fig. 4 to represent the nonlinear dynamics and the microwave linewidth are used in Fig. 6. The red dotted curve shows the maximum P1-frequency power for a specified bias current. Figure 6(a) shows the 3-dB microwave linewidth of the P1 oscillation when the field noise of the master laser is considered by setting Δv_M = 50 Hz while that of the slave laser is suppressed by setting Δv_S = 0 Hz. The narrowest microwave linewidth of the P1 oscillation reaches 1 MHz at the operating point of ($\tilde{J}^{S}$, f) = (1.8, 9 GHz), which falls on the curve of maximum P1-frequency power shown as the red dotted curve in Fig. 6(a). The curve of the maximum P1- oscillation power is close to the operating points where the P1 oscillation undergoes a frequency minimum with respect to the detuning frequency. The microwave linewidth of the P1 oscillation broadens as the detuning frequency is increased. An increase in the bias current of the slave laser narrows the microwave linewidth of the P1 oscillation, reaching a minimum-linewidth operating point. Further increase in the bias current of the slave laser beyond the minimum-linewidth operating point broadens the microwave linewidth of the P1 oscillation. Figure 6(b) shows the microwave linewidth of the P1 oscillation when the field noise of the slave laser is considered with Δv_S = 50 MHz for a normalized slave-laser bias current of $\tilde{J}^{S}$ = 1.222, while the field noise of the master laser is suppressed with Δv_M = 0 Hz. The narrowest linewidth of the P1 oscillation reaches 3 MHz at the operating point of ($\tilde{J}^{S}$, f) = (2, 12 GHz), which falls on the curve of maximum P1-frequency power shown as the red dotted curve in Fig. 6(b). Unlike the case when only the field noise of the master laser is considered, a broader operating region of narrow microwave linewidth of P1 oscillation is observed, as illustrated in Fig. 6(b). Figure 6(c) shows the microwave linewidth of the P1 oscillation when both the field noise of the slave laser and that of the master laser are considered with Δv_S = 50 MHz for a normalized slave-laser bias current of $\tilde{J}^{S}$ = 1.222 and Δv_M = 50 Hz. The narrowest microwave linewidth of the P1 oscillation reaches 7 MHz at the operating point of ($\tilde{J}^{S}$, f) = (2.2, 11 GHz), which falls on the maximum P1-frequency power shown as the red dotted curve in Fig. 6(c). Similar trends are observed, as compared to Fig. 6(a), in the microwave linewidth of the P1 oscillation as the detuning frequency or the bias current of the slave laser is increased beyond the narrowest-linewidth operating point, as illustrated in Fig. 6(c).

 

Fig. 6. Mapping of the 3-dB microwave linewidth Δf₀ of the P1 oscillation induced by an optically injected semiconductor laser as a function of the bias current of the slave laser and the detuning frequency while the injection strength is fixed at ξ = 0.15. (a) Only the master laser noise is considered, with Δv_M = 50 MHz and Δv_S = 0 Hz; (b) only the slave laser noise is considered, with Δv_S = 50 MHz and Δv_M = 0 Hz; (c) both slave laser noise and master laser noise are considered, with Δv_S = 50 MHz and Δv_M = 50 MHz. The P1 frequency is represented by thick contour curves in GHz, whereas the colored regions show the 3-dB microwave linewidth of the P1 oscillation in MHz with the yellow curves indicating a 3-dB microwave linewidth equal to 20 MHz. Uncolored regions represent complex dynamics and stably injection-locked fixed-point states, which are separated from the P1 dynamics by dense curves for the period-doubling bifurcation line and the Hopf bifurcation line, respectively. The red dotted curve shows the maximum P1-frequency power for a specified bias current.

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Figures 4–6 illustrate that the field noise of the master laser has a significant effect on the optical and microwave linewidths of the slave laser; it considerably broadens the microwave linewidth of the P1 oscillation. The contribution from the field of the slave laser or the master laser alone shows similar characteristics in the linewidth of the P1 oscillation. Under certain optimum operating conditions, the microwave linewidth of the P1 oscillation can be significantly minimized even when the contributions to the line broadening from the field noise of both the master and the slave lasers are considered, reaching linewidths that are an order of magnitude narrower than the optical linewidths of the master and slave lasers by properly adjusting the operating conditions of the optical injection system. The operating points of low sensitivity to the fluctuations in the detuning frequency are found in the operating regions where the frequency of the P1 oscillation undergoes a minimum with respect to the detuning frequency. These operating points fall in the regions where the P1-frequency reaches a maximum power while having the narrowest microwave linewidth, as illustrated in Figs. 4 and 6. The operating points of low sensitivity to the fluctuations in the injection strength are found in the regions where the frequency of the P1 oscillation undergoes a minimum with respect to the injection strength. These operating points show narrowed microwave linewidths for the P1 oscillation although the narrowest microwave linewidth is found in the regions where the power of the P1 oscillation is at a maximum, as illustrated in Fig. 5. The operating points of low sensitivity to the fluctuations in the bias current of the slave laser are found in regions where the frequency of the P1 oscillation undergoes a minimum with respect to the bias current. These operating points fall in the regions where the power of the P1 oscillation is at a maximum and the microwave linewidth of the P1 oscillation is the narrowest, as illustrated in Figs. 5 and 6.

6. Conclusion

The stability of the P1 oscillations is disturbed by the intrinsic laser noise of the master and slave lasers along with the time-dependent fluctuations in the operating parameters of the optically injected semiconductor laser system. The stability of the P1 oscillation is investigated through the microwave linewidth of the P1 oscillation when the intrinsic laser noise sources of the master and slave lasers are systematically considered. Optical injection generates P1 oscillations that have microwave linewidths narrower than the optical linewidths of the free-running master and slave lasers. The narrow microwave linewidth is attributed to the phase correlation of the optical frequency components of the P1 dynamics. The phase correlation of the optical frequency components degrades as the operational parameters are increased due to reduced coupling between the refractive index of the gain medium and the optical gain, resulting in a broad microwave linewidth for the P1 oscillation. Unlike the carrier noise of the slave laser, which has a negligible effect on the microwave linewidth of the P1 oscillation, the field noise of the master laser contributes significantly to the microwave linewidth of the P1 oscillation and cannot be neglected. The linewidth of the regeneratively amplified optical frequency component is solely determined by the free-running optical linewidth of the master laser. On the other hand, the optical linewidth of the principal oscillation frequency component depends on the free-running optical linewidths of both the master and the slave lasers. The field noise of both lasers broadens the optical linewidths of the master and slave lasers, which in turn broaden the microwave linewidth of the P1 oscillation. The general trend of the microwave linewidth of the P1 oscillation is the same when the field noise strength is increased, showing a microwave linewidth of the P1 oscillation that is an order of magnitude narrower than the optical linewidths of the master and slave lasers in operating regions of a high P1-frequency microwave power. Operating points that have low sensitivity to the fluctuations in an operating parameter are situated in a region where the P1 oscillation undergoes a frequency minimum with respect to the corresponding operating parameter. Low-sensitivity operating points show narrow microwave linewidths of the P1 oscillation although not necessarily the narrowest microwave linewidth possible.

The characteristics described here of the microwave linewidth of the P1 oscillation complement the continuing efforts to stabilize the P1-oscillation frequency [24,44]. They reveal that narrow microwave linewidths beyond those reached at low-sensitivity points are possible by further adjusting the operating parameters. Furthermore, these characteristics show the capability of utilizing P1 oscillations of optically injected semiconductor lasers in optical communication, signal processing, and optical ranging applications.