1. Introduction

Semiconductor lasers subject to an external optical injection can generate limit-cycle dynamics of period-one (P1) and period-two (P2) oscillations that are resonant at frequencies reaching six times the relaxation resonance of the solitary laser [1–3]. These oscillations are generated without the need for bandwidth-limiting electronic devices and are broadly tunable with single-sideband characteristics [4,5]. Furthermore, recent developments showed that the P1 and P2 dynamics have inherent low sensitivity (LS) to fluctuations in the intensity or frequency of both lasers when properly perturbed [6–9]. The LS operating points emerge due to destructive interference of many frequency components of the optical spectrum through wave-mixing at a local P1- or P2-frequency extremum with respect to one of the operational parameters. These characteristics of the periodic dynamics generated by optically injected lasers made them attractive for many applications including radio-over-fiber communication, all-optical signal conditioning and processing, lidar and velocity measurements [10–14].

Under optical injection, the laser cavity configuration and the intrinsic properties of the semiconductor medium play a crucial role in the generated nonlinear dynamics and their underlying characteristics [15–23]. This suggests that periodic dynamics at operating points with low sensitivity to fluctuations in the operational parameters can be enhanced depending on the semiconductor laser under use. Figure 1 shows a schematic of an optical injection setup. A master laser (ML), through an external modulator, optically injects a slave laser that is connected to a radio-frequency (RF) synthesizer. The generated P1 and P2 oscillations can be tuned by varying the bias current of the slave laser, J̃, detuning frequency between the master and slave lasers, f, or the injection strength, ξ. To identify P1 dynamics with low sensitivity to fluctuations in the bias current of the slave laser (LS_J͂), a small-signal low-frequency bias-current modulation is applied to the slave laser. The LS_J͂ points can be found at a local P1 extremum with respect to the bias current as well as at a P1 frequency extremum with respect to the detuning frequency. The latter makes LS_J͂ points insensitive to fluctuations in the temperature and bias current of the slave laser since the detuning frequency is controlled by the temperature of the two lasers. P1 dynamics with low sensitivity to frequency fluctuation (LS_f) are identified through an applied small-signal low-frequency modulation to the detuning frequency. The LS_f operating points suppress frequency fluctuations emerging from temperature and bias-current variations in the master and slave lasers. The LS_f points arise at a local P1 extremum with respect to the detuning frequency. Similarly, P1 dynamics with low sensitivity to intensity fluctuation (LS_ξ) are identified through an applied small-signal low-frequency modulation to the injection strength. The LS_ξ operating points suppress intensity fluctuations emerging from temperature and bias-current variations in the master and slave lasers. The LS_ξ points arise at a local P1 extremum with respect to the injection strength. The stability of the P1 dynamic is measured through the amplitude of the emerging modulation sidebands relative to the P1-frequency amplitude, in dBc. This measure is used in order to be able to quantify the contribution of a specified fluctuating operating parameter on the stability of the P1 oscillation frequency. The inset of Fig. 1 is a representative LS operating point at a P1-frequency minimum (black curve) with respect to the detuning frequency. The positive (blue) and negative (red) modulation sideband responses relative to the P1-frequency carrier are suppressed at the P1-frequency minimum when detuning-frequency modulation is applied to the master laser output, indicting an LS_f operating point.

 

Fig. 1 A schematic of the optical injection setup with external modulation on the master laser (ML) and direct modulation on the slave laser (SL). MOD: amplitude/frequency modulator; RF: radio frequency synthesizer. The modulation indices m_J̃, m_ξ, and m_f, and the modulation frequencies f_J̃, f_ξ, and f_f, represent modulation to the bias current, injection strength, and detuning frequency, respectively. The inset shows a representative LS_f point where the responses of the positive (blue) and negative (red) modulation sidebands show dips around a local minimum of the P1 frequency (black) with respect to the detuning frequency at a fixed ξ = 0.08 and J̃ = 1.222. The progression of the negative modulation sideband response as the modulation frequency is increased from 100 MHz to 2 GHz at a fixed modulation index m_f = 0.8 is shown in the red curves, shifted by – 30 dB for clarity. The progression of the negative modulation sideband response as the modulation index is increased from 0.4 to 2 at fixed a modulation frequency of f_f = 500 MHz is shown in the upper red curves.

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To better understand the role of the intrinsic parameters of the semiconductor laser on the LS operating points, two-dimensional maps with respect to the three operational parameters of the optically injected laser are calculated showing the LS_J̃, LS_f, and LS_ξ operating points along with the P1 and P2 frequencies. The three types of low-sensitivity operating points can be found at or close to a local extremum of the P1 or P2 frequency with respect to the corresponding operational parameter. Only the negative modulation sideband response is plotted for each operational-parameter modulation, since the positive and negative modulation sidebands have relatively the same response at low frequency, as shown in the inset of Fig. 1. For each variation of the intrinsic parameters, three maps are calculated by fixing one of the three operational parameters J̃, f, or ξ while varying the other two. A P1 dynamic operating point is considered a LS operating point if the corresponding modulation sideband response relative to the P1-frequency amplitude is below −30 dBc. Unlike previous work [15,16], only the P1 and P2 dynamics at the highly nonlinear region, where LS regions exist, is of interest.

2. Numerical model

The dynamics of a single-mode optically injected semiconductor laser under either direct modulation on the slave laser or external modulation on the master laser is described by the intracavity optical field amplitude A(t) and the charge carrier density N(t). For the free-running laser biased above threshold at an injection current density J, the laser oscillates at an optical frequency ω₀ with a steady-state field amplitude A₀ and a steady-state carrier density N₀. The optical injection is described by an optical field amplitude A_i with an optical frequency ω_i [9].

For numerical calculations, these quantities are recast with respect to the amplitude and carrier density of the steady-state, free-running slave laser condition as a_r + ia_i = A/|A₀| and 1 + n͂ = N/N₀. Here, a_r and a_i represent the real and imaginary parts of the normalized complex field amplitude, respectively, and ñ is the normalized carrier density. The optical injection is characterized by the detuning frequency f = (ω_i – ω₀)/2π and the normalized dimensionless injection strength ξ = η|A_i|/γ_c|A₀|, where η is the injection coupling rate and γ_c is the cold cavity decay rate. The normalized bias current above threshold is given by J͂ = (J/ed – γ_sN₀)/γ_sN₀, where e, d, and γ_s are the electronic charge, active layer thickness, and spontaneous carrier relaxation rate, respectively. The rate equations become [9,17]:

The stability of P1 oscillation frequency induced by an optically injected semiconductor laser is influenced by the fluctuations in the operating parameters as well as the random disturbances from spontaneous emission noise and charge-carrier noise in the slave laser. While the effects of bias-current, detuning-frequency, and injection-strength fluctuations are suppressed at the LS_J̃, LS_f, and LS_ξ, respectively, spontaneous emission noise and charge-carrier noise still play a role. In the model used (1)-(3), fluctuating source terms with contributions to both the optical-field and carrier-density equations could be introduced to account for spontaneous emission. At an LS_J̃ point, the carrier-density equation contribution is eliminated, yet the field-equation contribution still exists. Similarly, at a LS_f, or LS_ξ operating point only frequency, or amplitude contribution of spontaneous emission in the field-equation is eliminated, respectively. Therefore, the noise source terms in the optical-field and carrier-density equations are neglected in the model so that contribution from a specific operational-parameter fluctuation is solely considered.

To compare the theoretical and experimental results, a 500 MHz modulation frequency was chosen to easily identify the modulation sideband responses in the optical and power spectra, as shown in a previous effort [9]. Moreover, using a 500 MHz modulation frequency in our numerical simulations allows direct comparison against previous work [8]. Therefore, to mimic the various operating-parameter fluctuations a weak modulation signal is added at f_J̃ = f_ξ = f_f = 500 MHz for all the generated spectra. Noting that for lower modulation frequencies the modulation sideband responses show deeper dips [7]. As the modulation frequency is increased, the modulation sideband dips gradually fade away [8]. This is demonstrated in the inset of Fig. 1, the red curves show the progression of the negative modulation sideband response as the modulation frequency is increased from 100 MHz to 2 GHz. On the other hand, the increase in the modulation amplitude results an increase in the modulation sideband amplitudes, therefore, shifting the response of the modulation sidebands progressively to higher relative amplitudes. This is demonstrated in the inset of Fig. 1, the upper red curves show the progression of the negative modulation sideband response as the modulation index is increased from 0.4 to 2 at an LS_f operating point.

The laser intrinsic parameters have been experimentally extracted for various lasers [24–28] and the rate equation model used here has been demonstrated to generate all the experimentally observed phenomena. For lasers with different intrinsic parameters, a dual-beam optically injected semiconductor laser showed qualitative agreement between theory [29] and experiment [30]. More recently, the spectral properties at a LS operating point of an optically injected laser showed excellent quantitative agreement [9] when the experimentally extracted laser parameters are used in the theoretical model. As a basis the initial intrinsic parameters adopted here, which have been previously mapped out [8], are for a InGaAsP/InP DFB semiconductor laser with γ_c = 5.36 × 10¹¹ s⁻¹, γ_s = 5.96 × 10⁹ s⁻¹, γ_n = J̃ • 6.162 × 10⁹ s⁻¹, γ_p = J̃ • 1.563 × 10¹⁰ s⁻¹, b = 3.2, and bʹ = 3.2 having a relaxation resonance of f_r = 10.25 GHz at J̃ = 1.222 [25]. By adjusting the intrinsic laser parameters, the model used here can effectively describe a variety of laser cavity configurations, such as Fabry-Perot [31], DFB [28], VCSEL [27], etc., with different gain structures (heterostructure, quantum-well) and numerous semiconductor materials. The model used here does not account for the effects of the quantum-confined carriers and therefore does not reflect the dynamical characteristics of nanostructured lasers, such as quantum-dot and quantum-dash devices [19]. Nonetheless, it has been demonstrated experimentally that LS operating points do exist in an optically injected quantum-dot lasers [23].

3. Results and discussion

3.1 Effects of the cold-cavity decay rate on the low-sensitivity operating points

The cold-cavity decay rate or photon lifetime of the passive cavity represents the decay rate of the optical energy stored in the cavity and varies for different cavity configurations [32]. To reflect the effects of the cold-cavity decay rate on the LS operating points when the optically-injected laser is operated in the P1 dynamic, γ_c is varied from γ_c = 3 × 10¹¹ s⁻¹ to γ_c = 8 × 10¹¹ s⁻¹ while the other laser parameters are fixed to their experimentally defined values. This change in the cavity-decay rate generates a laser with a relaxation resonance of f_r = 7.75 GHz and f_r = 12.47 GHz at J͂ = 1.222, respectively. Figure 2 shows mapping of the nonlinear dynamics induced by a semiconductor laser subject to an external optical injection where the P1-dynamic region (black curves, in GHz) and the stable-locking region (uncolored) are separated by the Hopf-bifurcation line (thick black curve). Chaotic (black areas) and P2 dynamics are separated by period-doubling bifurcation (dense curve). The saddle-node bifurcation (without thick or dense black curves) separate the P1 dynamic from the stable-locking region. The colored regions show contour curves representing the amount of suppression of the negative modulation sideband response relative to the P1-frequency amplitude when the suppression is at –30 dBc or below. A bias-current, detuning-frequency, or injection-strength modulation response of –30 dBc or below, indicate LS_J͂ (Green), LS_f (blue), or LS_ξ (red), operating points, respectively. An operating point that has both LS_J͂ and LS_f (light blue), LS_J͂ and LS_ξ (yellow), LS_f and LS_ξ (magenta) show simultaneous insensitivity to fluctuation in two of the three operating parameters. Optimal operating points (gray) show simultaneous insensitivity to fluctuation in all three operating parameters. The Venn diagram, inset in Fig. 2(a-i), shows the respective color codes for the various LS operating points and the corresponding colors for simultaneous LS operating points.

 

Fig. 2 Mapping of the LS regions for (a) γ_c = 3 × 10¹¹ s⁻¹ and (b) γ_c = 8 × 10¹¹ s⁻¹ while the other laser parameters are fixed to their experimentally defined values. The maps are plotted as a function of the injection strength and detuning frequency for fixed J̃ = 1.222 in column (i), as a function of the injection strength and bias current for fixed f = 10 GHz in column (ii), and as a function of the detuning frequency and bias current for fixed ξ = 0.15 in column (iii). The black contour curves represent a constant P1 and P2 frequency, in GHz, and separate the stable locking (uncolored region) from chaotic dynamics (black regions). P1 dynamics and P2 dynamics are separated by the period-doubling bifurcation (dense curve). The contour curves in the colored regions show the amount of suppression of the negative modulation sideband when the suppression is −30 dBc or below, for the various LS operating points. The Venn diagram shows the color coding used for individual and multiple LS operating points.

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Rows (a) and (b) in Fig. 2 show the LS regions for γ_c = 3 × 10¹¹ s⁻¹ and γ_c = 8 × 10¹¹ s⁻¹, respectively. Columns (i), (ii), and (iii) in Fig. 2 show the LS regions when the bias current is fixed at J̃ = 1.222, detuning frequency is fixed at f = 10 GHz, and injection strength is fixed at ξ = 0.15, respectively, while the other two operational parameters are varied. The general effect of γ_c is to destabilize the system as it is increased, shifting the dynamics to high detuning frequencies and lower injection strengths. The shift of the complex dynamics is due to an increase in the relaxation resonance frequency, but not the damping rate, of the laser as γ_c is increased. The highly nonlinear region, bounded between the Hopf-bifurcation line and the chaotic region, widens as γ_c is increased. For large γ_c values, broader P1-frequency extrema with respect to the detuning frequency and the bias current is observed while narrower P1-frequency extrema with respect to the injection strength is observed, as demonstrated in Fig. 2(b). For the same injection strengths higher P1 frequencies are achieved for larger γ_c values, as demonstrated by comparing the same operating points in column (i), (ii) or (iii) of Fig. 2. The broad P1-frequency extrema with respect to J̃ and f create wider LS_J̃ and LS_f regions, respectively, as demonstrated by the wide colored regions in Fig. 2(b) as opposed to Fig. 2(a). On the other hand, for large γ_c values, the LS_ξ regions are reduced significantly due to narrower P1 extrema with respect to ξ. This is not the case, however, in the highly nonlinear region where LS_ξ regions are enhanced as demonstrated in column (i) and (ii) of Fig. 2, where an LS_ξ operating points at a P1 frequency of 12 GHz is increased to 26 GHz as γ_c is increased to 8 × 10¹¹ s⁻¹. Consequently, for large γ_c values, simultaneous LS_J͂ and LS_f regions are enhanced, indicated by the light blue region in Fig. 2(b), while simultaneous LS_J͂ and LS_ξ are reduced indicated by the yellow regions in Fig. 2(b). It is expected that LS_J͂ cover larger regions than LS_f or LS_ξ regions since LS_J͂ operating points are found at P1 extrema with respect to both J̃ and f [6,7]. An optimal operating point, demonstrated previously [8], is observed close to where the Hopf-bifurcation line and the saddle-node bifurcation line meet, gray dot in Fig. 2(a-ii). Further increase or decrease of the γ_c value beyond the values in Fig. 2 generate the same general behavior on the LS regions described here. Therefore, for enhanced LS_J͂ and LS_f regions large γ_c value is favored, on the other hand, for enhanced LS_ξ regions a small γ_c value is favored.

3.2 Effects of the spontaneous carrier decay rate on the low-sensitivity operating points

The spontaneous carrier decay rate represents the rate of carrier recombination contributed from radiative and nonradiative spontaneous emission and is constant regardless of the laser power. For an optically injected semiconductor laser a change in the spontaneous carrier decay rate from γ_s = 1 × 10⁹ s⁻¹ to γ_s = 10 × 10⁹ s⁻¹ showed only marginal change in the nonlinear dynamics with slight variation in the P1 frequency of a few GHz for the same operational parameters. This is expected since the effects of spontaneous emission recombination rate are dominated by stimulated emission recombination rate in the optically injected laser as has been demonstrated previously [15]. Therefore, γ_s does not show any significant effects on the LS regions and the results are not presented here.

3.3 Effects of the differential carrier relaxation rate on the low-sensitivity operating points

The differential carrier relaxation rate represents the dependence of the gain parameter on the carrier density under dynamical perturbation and is proportional to the intracavity photon density [32]. Rows (a) and (b) in Fig. 3 show the LS regions for γ_n = J̃ • 3 × 10⁹ s⁻¹ and γ_n = J̃ • 8 × 10⁹ s⁻¹, respectively. This change in the differential carrier relaxation rate generates a laser with a relaxation resonance of f_r = 7.26 GHz and f_r = 11.65 GHz at J͂ = 1.222, respectively. Columns (i), (ii), and (iii) in Fig. 3 show the LS regions when the bias current is fixed at J̃ = 1.222, detuning frequency is fixed at f = 10 GHz, and injection strength is fixed at ξ = 0.15, respectively, while the other two operational parameters are varied. Similar convention to represent the nonlinear dynamics and the various LS regions used in Fig. 2 are used in Fig. 3. It has been demonstrated that for an optically injected semiconductor laser a large γ_n value destabilizes the system [15]. Comparing Fig. 3(a) and 3(b) shows the destabilizing effect through an expansion of the P1 dynamic and chaotic regions and the emergence of a P2 dynamic region. This is expected due to the increase in the relaxation resonance frequency as γ_n is increased. No change in the saddle-node bifurcation region is observed. For the same operating point, a higher P1 frequency is realized for lasers with a larger γ_n value, as demonstrated by comparing the same operating points in column (i), (ii) or (iii) of Fig. 3. The LS_J͂, LS_f, LS_ξ regions expand for large γ_n values, showing simultaneous LS_J͂ and LS_f at high P1 frequencies. This is demonstrated by comparing Figs. 3(a-i) to 3(a-ii) and Figs. 3(b-i) to 3(b-ii). The LS_ξ regions close to the emerging P2 dynamic are suppressed as illustrated in rows (a) and (b) of Fig. 3. The expansion in the LS regions are due to the expansion in the P1 and P2 regions. Further increase or decrease of the γ_n value beyond the values in Fig. 3 generate the same general behavior on the LS regions described here. Therefore, for stable, high-frequency P1 oscillations a semiconductor laser with a large γ_n value is favored.

 

Fig. 3 Mapping of the LS regions for (a) γ_n = J͂ • 3 × 10⁹ s⁻¹ and (b) γ_n = J͂ • 8 × 10⁹ s⁻¹ while the other laser parameters are fixed to their experimentally defined values. The maps are plotted as a function of the injection strength and detuning frequency for fixed J̃ = 1.222 in column (i), as a function of the injection strength and bias current for fixed f = 10 GHz in column (ii), and as a function of the detuning frequency and bias current for fixed ξ = 0.15 in column (iii). The black contour curves represent a constant P1 and P2 frequency, in GHz, and separate the stable locking (uncolored region) from chaotic dynamics (black regions). P1 dynamics and P2 dynamics are separated by the period-doubling bifurcation (dense curve). The contour curves in the colored regions show the amount of suppression of the negative modulation sideband when the suppression is −30 dBc or below, for the various LS operating points. The Venn diagram shows the color coding used for individual and multiple LS operating points.

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3.4 Effects of the nonlinear carrier relaxation rate on the low-sensitivity operating points

The nonlinear carrier relaxation rate represents the dependence of the gain parameter on the intracavity photons under dynamical perturbation and is proportional to the intracavity photon density. It is responsible for gain compression due to saturation effects such as hole burning, dynamic carrier heating, and population pulsations [32].

Rows (a) and (b) in Fig. 4 show the LS regions for γ_p = J̃ • 1 × 10¹⁰ s⁻¹ and γ_p = J̃ • 3 × 10¹⁰ s⁻¹, respectively. This change in the nonlinear carrier relaxation rate generates a laser with a relaxation resonance of f_r = 10.2 GHz and f_r = 10.38 GHz at J͂ = 1.222, respectively. Columns (i), (ii), and (iii) in Fig. 4 show the LS regions when the bias current is fixed at J̃ = 1.222, detuning frequency is fixed at f = 10 GHz, and injection strength is fixed at ξ = 0.15, respectively, while the other two operational parameters are varied. Similar convention used to represent the nonlinear dynamics and the various LS regions in Fig. 2 are used in Fig. 4. It has been demonstrated that for an optically injected semiconductor laser a large γ_p value stabilizes the system [15]. Figure 4(a) and 4(b) display the stabilizing effect of a large γ_p value through an expansion of the stable-locking region and reduction in the P1 and chaotic dynamic regions whereas the P2 region vanishes completely. The shift of the dynamics to higher frequencies is attributed to an increase in the relaxation resonance frequency whereas the stabilization is attributed to an increase in the damping rate of the laser. Clearly, γ_p has a larger effect on the overall dynamics as compared to γ_n. Similar to γ_n, no change in the saddle-node bifurcation region is observed as γ_p is varied. For the same operating points in the P1 dynamic region, only a slight increase in the P1 frequency is achieved for smaller γ_p values, as demonstrated by comparing the same operating points in column (i), (ii) or (iii) of Fig. 4. The LS_J͂ and LS_f regions expand for small γ_p values, showing simultaneous LS_J͂ and LS_f at high P1 frequencies. This is demonstrated by comparing the green, blue, and light blue regions of Fig. 4. The highly nonlinear region expands with small values of γ_p, generating larger regions of LS_ξ operation away from chaotic regions as demonstrated in Figs. 4(a-i) and 4(a-ii). The enlargement in the LS regions are due to the expansion in the P1 and P2 regions associated with the destabilizing effect of a small γ_p value. Further increase or decrease of γ_p value beyond the values in Fig. 4 generate the same general behavior on the LS regions described here. Therefore, for stable, high-frequency P1 oscillations a semiconductor laser with a small γ_p value is favored. Setting γ_p = 0 the LS regions still exist with the same general characteristics described above for a small γ_p value, although doing so showed substantial deviation from the experimental extracted values, as has been demonstrated previously [9].

 

Fig. 4 Mapping of the LS regions for (a) γ_p = J͂ • 1 × 10¹⁰ s⁻¹ and (b) γ_p = J͂ • 3 × 10¹⁰ s⁻¹ while the other laser parameters are fixed to their experimentally defined values. The maps are plotted as a function of the injection strength and detuning frequency for fixed J̃ = 1.222 in column (i), as a function of the injection strength and bias current for fixed f = 10 GHz in column (ii), and as a function of the detuning frequency and bias current for fixed ξ = 0.15 in column (iii). The black contour curves represent a constant P1 and P2 frequency, in GHz, and separate the stable locking (uncolored region) from chaotic dynamics (black regions). P1 dynamics and P2 dynamics are separated by the period-doubling bifurcation (dense curve). The contour curves in the colored regions show the amount of suppression of the negative modulation sideband when the suppression is −30 dBc or below, for the various LS operating points. The Venn diagram shows the color coding used for individual and multiple LS operating points.

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3.5 Effects of the linewidth enhancement factor on the low-sensitivity operating points

In semiconductor lasers, the linewidth enhancement factor is the ratio of the carrier-induced changes of the real and imaginary parts of the complex susceptibility. Through its value, it represents the strength of the coupling between the phase and amplitude of the optical field and is responsible for the emergence of instabilities in an optically injected semiconductor laser [16].

Rows (a) and (b) in Fig. 5 show the LS regions for b = 1 and b = 4, respectively. Columns (i), (ii), and (iii) in Fig. 5 show the LS regions when the bias current is fixed at J̃ = 1.222, detuning frequency is fixed at f = 10 GHz, and injection strength is fixed at ξ = 0.15, respectively, while the other two operational parameters are varied. Similar convention used to represent the nonlinear dynamics and the various LS regions in Fig. 2 are used in Fig. 5. Comparing Fig. 5(a) and 5(b) shows that semiconductor lasers with a small value of b stabilizes the system. The P1 dynamic region induced by Hopf-bifurcation shrinks, whereas, the P1 dynamic region induced by saddle-node bifurcation expands, as b is reduced. Meanwhile, the chaotic and P2 dynamic regions vanish completely removing the highly nonlinear region between the Hopf-bifurcation line and chaos, as illustrated in Fig. 5(a) and 5(b). Higher P1 frequencies are generated for larger b values for the same operating parameters, as demonstrated by comparing the same operating points in column (i), (ii) or (iii) of Fig. 5. The LS_J̃ and LS_ξ regions are reduced for a large b value, as demonstrated in the smaller green, red, and yellow regions in Fig. 5(b) as opposed to Fig. 5(a). On the other hand, the LS_f regions are enhanced for a large b value, as demonstrated in the larger blue regions of Fig. 5(b) as opposed to Fig. 5(a). Figure 5(a) shows that for small b values the LS_J͂ regions completely overlap the LS_f regions. Except at high bias currents, the P1 region shows complete stability to at least one of the three operating parameters. Figure 5(b) shows that as the b value is increased, the P1 region expands generating LS regions that are more localized close to the P1-frequency extrema, therefore, reducing the simultaneous LS regions shown in Fig. 5(a). Varying b beyond the values chosen in Fig. 5, show the same general characteristics described above. Therefore, for stable, high-frequency P1 oscillations a semiconductor laser with a large b value is favored. On the other hand, for stable P1 frequencies to multiple operating parameter fluctuations, a semiconductor laser with a small b value is favored.

 

Fig. 5 Mapping of the LS regions for (a) b = 1 and (b) b = 4 while the other laser parameters are fixed to their experimentally defined values. The maps are plotted as a function of the injection strength and detuning frequency for fixed J̃ = 1.222 in column (i), as a function of the injection strength and bias current for fixed f = 10 GHz in column (ii), and as a function of the detuning frequency and bias current for fixed ξ = 0.15 in column (iii). The black contour curves represent a constant P1 and P2 frequency, in GHz, and separate the stable locking (uncolored region) from chaotic dynamics (black regions). P1 dynamics and P2 dynamics are separated by the period-doubling bifurcation (dense curve). The contour curves in the colored regions show the amount of suppression of the negative modulation sideband when the suppression is −30 dBc or below, for the various LS operating points. The Venn diagram shows the color coding used for individual and multiple LS operating points.

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3.6 Effects of the gain saturation factor on the low-sensitivity operating points

In semiconductor lasers, the gain saturation factor is the ratio of the real and imaginary parts of the complex susceptibility associated with changes in the circulating optical power. It takes a positive, negative, or zero value when the laser is operating on the low-frequency side, high-frequency side, or coincides with the gain peak, respectively [17,28]. It has been introduced to account for the asymmetric modulation response in semiconductor lasers under direct current modulation [6,9,17,33,34]. Together with Fig. 5(b), Fig. 6 shows mapping of the LS regions as the gain saturation factor is reduced from b = bʹ = 4 in Fig. 5(b) to b = 4, bʹ = 0 in Fig. 6(a) and b = 4, bʹ = − 4 in Fig. 6(b) while the other laser parameters are fixed to their experimentally defined values.

 

Fig. 6 Mapping of the LS regions for b = 4 (a) bʹ = 0 and (b) bʹ = − 4 while the other laser parameters are fixed to their experimentally defined values The maps are plotted as a function of the injection strength and detuning frequency for fixed J̃ = 1.222 in column (i), as a function of the injection strength and bias current for fixed f = 10 GHz in column (ii), and as a function of the detuning frequency and bias current for fixed ξ = 0.15 in column (iii). The black contour curves represent a constant P1 and P2 frequency, in GHz, and separate the stable locking (uncolored region) from chaotic dynamics (black regions). P1 dynamics and P2 dynamics are separated by the period-doubling bifurcation (dense curve). The contour curves in the colored regions show the amount of suppression of the negative modulation sideband when the suppression is −30 dBc or below, for the various LS operating points. The Venn diagram shows the color coding used for individual and multiple LS operating points.

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Columns (i), (ii), and (iii) in Fig. 6 show the LS regions when the bias current is fixed at J̃ = 1.222, detuning frequency is fixed at f = 10 GHz, and injection strength is fixed at ξ = 0.15, respectively, while the other two operational parameters are varied. Similar convention used to represent the nonlinear dynamics and the various LS regions in Fig. 2 are used in Fig. 6. It has been demonstrated that a negative value of bʹ destabilizes the system [17], this is more evident for a laser with large values of b, as illustrated in Fig. 6. Comparing Fig. 5(b) to Figs. 6(a) and 6(b) shows considerable enhancement in the P1, P2, and chaotic regions as bʹ is reduced. The constant P1- and P2-frequency contour curves show emerging extrema with respect to the operational parameters that are not found for positive values of bʹ. Unlike the effect of b variation on the P1 frequency, only minimal increase in P1 frequency for the same operational parameters is achieved as demonstrated by comparing the same operating points in column (i), (ii) or (iii) of Fig. 6. The P1-dynamic LS_J̃ regions show significant reduction as bʹ is reduced, removing the simultaneous LS_J̃ and LS_ξ at high detuning frequencies and high bias currents, as indicated by the reduction in the green and yellow regions in Fig. 6 as opposed to Fig. 5(b). The LS_f regions are more localized at proximity to the P1- or P2-frequency extrema, as bʹ is reduced to a more negative value. Except for the new emerging P1 and P2 extrema, the LS_ξ are relatively unaffected by the variation of the gain saturation factor from bʹ = 4 to bʹ = − 4, as demonstrated by comparing 5(b) to 6(a) and (b). The emerging P2-dynamic regions show overlapping LS_J̃, LS_f, and LS_ξ operating points where optimal operation due to external perturbation is achieved. Although the modulation sideband response is calculated around the P2 frequency, similar behavior is observed in the modulation sideband response around the P1 frequency. The general characteristics for a different laser with different values of b has the same general behavior as described above for positive and negative values of bʹ. Therefore, for optimal operating points at relatively low-frequency a laser with a negative bʹ value is favored. On the other hand, for broad P1-dynamic LS regions at high frequency, a laser with a positive bʹ value is favored.

3.7 Low-sensitivity operating points to multiple parameter fluctuations

The mapping of the low-sensitivity operating points, demonstrated through the modulation sideband response of the P1 oscillation, represents the stability of the P1 oscillation frequency against fluctuation in one of the operating parameters. The overlapping regions in the maps show the stability of the P1 oscillation against multiple parameter fluctuations when the operating-parameter modulation is applied individually. To further demonstrate the stability of the P1 oscillation against the fluctuation of multiple parameters, modulation is applied to two operating parameters simultaneously.

Figure 7(a) shows the negative modulation sideband responses when individual (dashed) and simultaneous (solid) modulation is applied near a local minimum of the P1 frequency (black) with respect to the injection strength at f = 10 GHz, J̃ = 1.222, b = 1, and bʹ = 1 while the other intrinsic parameters are fixed to their experimentally defined values. The individual modulation sideband responses when injection-strength or bias-current modulation is applied indicate a LS_ξ and LS_J͂, respectively. The overlapping modulation responses below −30dBc, corresponding to the yellow region in Fig. 5(a-i), indicate a low-sensitivity operating point with respect to both injection-strength and bias-current fluctuation. The modulation response when simultaneous injection-strength and bias-current modulation are applied also indicate a low-sensitivity point to multiple parameter fluctuations (LS_ξJ͂), demonstrated by the solid red line in Fig. 7(a). Figure 7(b) shows the negative modulation sideband responses when individual (dashed) and simultaneous (solid) modulation is applied near a local minimum of the P1 frequency (black) with respect to the detuning frequency at ξ = 0.08, J̃ = 1.222, and γ_p = J͂ • 1 × 10⁹ s⁻¹while the other intrinsic parameters are fixed to their experimentally defined values. The individual modulation sidebands when detuning-frequency or bias-current modulation is applied indicate a LS_f and LS_J͂, respectively. The overlapping modulation responses below −30dBc, corresponding to the light blue region in Fig. 4(a-i), indicate a low-sensitivity operating point with respect to both detuning-frequency and bias-current fluctuation. The modulation response when simultaneous detuning-frequency and bias-current modulation are applied also indicate a low-sensitivity point to multiple parameter fluctuations (LS_fJ͂). Figure 7(c) shows the negative modulation sideband responses when individual (dashed) and simultaneous (solid) modulation is applied near a local minimum of the P2 frequency (black) with respect to the bias current at ξ = 0.125, f = 10 GHz, b = 4, and bʹ = − 4 while the other intrinsic parameters are fixed to their experimentally defined values. The individual modulation sidebands when injection-strength or detuning-frequency modulation is applied indicate a LS_ξ and LS_f, respectively. The overlapping modulation responses below −30dBc, corresponding to the magenta region in Fig. 6(b-ii), indicate a low-sensitivity operating point with respect to both injection-strength and detuning-frequency fluctuation. The modulation response when simultaneous injection-strength and detuning-frequency modulation are applied also indicate a low-sensitivity point to multiple parameter fluctuations (LS_ξf).

 

Fig. 7 Negative modulation sideband responses when individual (dashed) and simultaneous (solid) modulation is applied near a local minimum of the P1 frequency (black) with respect to: (a) the injection strength at f = 10 GHz, J̃ = 1.222, b = 1, and bʹ = 1 while the other intrinsic parameters are fixed to their experimentally defined values; (b) the detuning frequency at ξ = 0.08, J̃ = 1.222, and γ_p = J͂ • 1 × 10⁹ s⁻¹ while the other intrinsic parameters are fixed to their experimentally defined values; (c) the bias current at ξ = 0.125, f = 10 GHz, b = 4, and bʹ = − 4 while the other intrinsic parameters are fixed to their experimentally defined values.

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The generated modulation sideband dips, at a low-sensitivity operating point with respect to multiple operating parameter fluctuations, when simultaneous modulation is applied shows the combined effect of the individual modulation sideband responses, but not as a superposition of the individual modulation responses. The generated simultaneous modulation sideband responses are slightly shifted and are less pronounced compared to the individual modulation sideband responses. It is beyond the scope of the current work to determine detailed characteristics of the modulation sideband responses generated by simultaneous modulation.

4. Conclusion

The intrinsic parameters of an optically injected laser play a crucial role in the emerging nonlinear dynamics and on the stability of the P1 and P2 oscillations against fluctuation in the operational parameters. The various LS operating regions are generated due to destructive interference of the many frequency components through nonlinear wave mixing at a local P1 or P2 extremum that minimizes the response of both the optical power and gain medium. The change in the relaxation rates changes the relaxation resonance and the damping rate of the laser which, in turn, modify the induced laser dynamics creating P1 and P2 extrema with respect to the operational parameters. The intrinsic parameters relating the gain to the refractive index through carrier and intensity variations dominate the characteristics of the P1 and P2 dynamic LS regions. Detailed mapping as functions of the three operational parameters of the optically injected laser J̃, ξ, and f demonstrated the effects of each intrinsic parameter on the P1- and P2-dynamic LS regions. The differential and nonlinear carrier relaxation rates have opposing effects on the general nonlinear dynamics of an optically injected laser when they are varied. A large value of γ_n increases the P1 and P2 dynamic regions creating high-frequency P1 and P2 oscillations with broader regions of low sensitivity to fluctuations in the operational parameter. On the other hand, a small value of γ_p increases the P1 and P2 dynamic regions creating high-frequency P1 and P2 oscillations with broader regions of low sensitivity to fluctuations in the operational parameter. Regardless of the γ_p and γ_n values, the various low-sensitivity regions to parameter fluctuations always exist indicating that such reduced sensitivity to perturbation does not stem from the differential and nonlinear gain effects. The cavity decay rate is an order of magnitude larger than the other relaxation rates of the laser and effects the relaxation resonance but not the damping rate of the laser when it is varied. Therefore, a large γ_c shifts the dynamics to low injection strengths and high detuning frequencies, generating wider regions of LS_J̃ and LS_f but narrower regions of LS_ξ. The linewidth enhancement factor and the gain saturation factor are the most critical parameters in generating P1 and P2 dynamics with low sensitivity to operational-parameter fluctuations. A large b value expands the P1 and P2 dynamic regions creating high-frequency oscillations with narrower, nonoverlapping regions of low sensitivity to operational-parameter fluctuations. A negative bʹ value considerably enhances P2 dynamic regions generating frequency extrema with respect to multiple control parameters, but it suppresses LS_J̃ operating points significantly. Consequently, for applications requiring high-frequency, stable P1 oscillations to a specified operational-parameter fluctuation, a semiconductor laser with a large b and positive bʹ value at a relatively high bias current is favored. In contrast, for localized P1 and P2 oscillations with reduced sensitivity to multiple operational parameter fluctuations a laser with a small b value or a negative bʹ value operating at relatively low bias currents is favored. The relaxation rates can further tailor the P1 oscillations for specific applications.

A few remarks should be given when comparing the dynamic mappings presented in Figs. 2-6. The normalized injection strength, ξ, depends on the cold-cavity decay rate and implicitly on the bias current, as discussed in Section 2. Therefore, the comparison should be made noting that the necessary injection laser amplitude to reach the same injection level, ξ, for the various lasers is different. Scaling the injection strength to account for the change in the cavity decay rate, in Fig. 2(a) and 2(b), shows less prominent differences in the two mappings. Similarly, scaling the detuning frequency in the generated maps to the relaxation resonance frequency of each laser, shows a less prominent shift in the dynamics to higher detuning frequencies. Nonetheless, the general dependence of the LS operating points on the intrinsic parameters is the same as presented in Figs. 2-6.

The laser intrinsic parameters that most dominate the low-sensitivity periodic-dynamic regions are γ_c, b, and bʹ. The b and bʹ values depend on the gain medium and the laser structure and can be modified for example, by shifting the emission wavelength of the laser from the gain peak or by changing the gain structure of the laser. On the other hand, the γ_c value depends on the cavity configuration and can be modified for example by adjusting the cavity length or facet reflectivities. The other relaxation rates can be varied by adjusting the applied current and operating temperature of the optically injected laser experimentally. Therefore, through the general characteristics described here, optimized lasers that show favorable reduced sensitivity to fluctuation in the master and slave laser operating conditions can be realized. Furthermore, this work compliments on ongoing efforts to produce narrow-linewidth, low phase-noise, frequency-tunable P1 oscillations for photonic microwave applications [35–41].