Influence of Radiative and Non-Radiative Recombination Lifetimes and Feedback Strength on the States and Relative Intensity Noise of Laser Diode

Abstract

A systematic treatment of the influence of the optical feedback (OFB) and the ratio of the radiative and the non-radiative recombination lifetime (τ_r/τ_nr) on the relative intensity noise (RIN) and the system dynamics of a semiconductor laser (SL) is presented. We found that the ratio τ_r/τ_nr causes a significant change in the intensity of noise, states, and route to chaos of such a system. Laser transit from a continuous wave (CW) to a periodic oscillation (PO) becomes faster in terms of the OFB strength by increasing the ratio τ_r/τ_nr. The route to chaos was identified in three distinct operating regions, namely, PO, period doubling (PD), and sub-harmonics (SH), which are dependent on the ratio τ_r/τ_nr, and injection current. In the route to chaos regime, the ratio τ_r/τ_nr triggers a slight shift in the frequency with reference to the frequency of the solitary laser. At upmost levels of the current, the highest value of the ratio τ_r/τ_nr stabilizes the laser and stimulates it to operate in CW or PO. In the strong OFB region, when the ratio τ_r/τ_nr increases, the chaotic operation changed to CW or PO operation and RIN is suppressed close to the quantum noise level.

1. Introduction

Semiconductor lasers have the competence to meet demands for coming generation high speed optical networks. Since their development in 1962, there have been numerous numbers of efforts from different disciplines towards designing and investigating the structure of solid-state semiconductors as building blocks for lasers with the motivation of their advantages. As far as their fascinating properties go, they are accepted as “the laser of the future”, due to their wide tunability, easy integration, very good spectral response as transmission sources, and robust output powers. Recently, it has been shown that SLs preferentially operate in the presence of external OFB. OFB in some circumstances can destroy the stability in laser operation in the form of chaos [1,2,3,4], coherence collapse [5], and bistability [6,7,8,9]. However, it has been shown theoretically and experimentally that OFB can be used as an external controller with many applications requiring narrow linewidth operation [10,11], e.g., coherent communication systems, mode stabilization [12], and reduction in modulation-induced frequency chirp [13,14]. Previous models dealing with OFB were applicable under some assumptions and approximations and simply incorporated the OFB by adding phenomenologically a time-delayed feedback term to the standard laser rate equations [15,16]. From a practical point of view, the external OFB is a technical and subtle problem and, therefore, it is important to have a detailed understanding of the behavior of a laser with OFB. This will include reliable analysis of SLs under an arbitrary amount of OFB beyond the regime of weak OFB, and the round trips of the laser light in both the laser cavity and the external cavity are counted as time delays of the laser light at the front facet, taking into consideration the transmission- and reflection-induced phase changes of the lasing field [17,18,19]. The dynamics of an SL with OFB in the low-frequency fluctuation regime were studied, with emphasis on the effects of single-mode and multimode regimes [20]. Nonlinearity dynamics of SLs in the presence of active OFB has been observed [21,22]. An improved theoretical model analyzing the dynamics and operation of SLs under OFB was given with the applicability for arbitrary strength of the feedback from weak to very strong [17].

However, the operation states and dynamics of the SL are influenced by the radiative and non-radiative recombination lifetime ratio τ_r/τ_nr, which has a strong effect on the threshold current, carrier lifetime, output saturation, turn-on time delay, and spontaneous emission factor [23,24,25]. A good understanding of the impact of the carrier recombination mechanisms in SLs on the dynamics and intensity noise is essential for designing SLs with high static and dynamic performance. In particular, it is important to maximize the rate of τ_r/τ_nr processes that increase the efficiency of these devices [25]. Abdulrhmann et al. investigated the influence of non-radiative recombination lifetime on the quantum noise of the 1550-nm InGaAsP/InP solitary laser in reference [26]. However, based on our knowledge, there is no investigation until now on the impact of the OFB strength and the ratio τ_r/τ_nr on the dynamics and RIN of SLs. In this paper, we propose a new numerical analysis of the influence of OFB, as well as the ratio of the radiative and the non-radiative recombination lifetime (τ_r/τ_nr), on the relative intensity noise and dynamics of SLs. We have shown and demonstrated that our results and findings allow for an on-demand, fast, and robust building block laser system, even in the presence of radiative and non-radiative interactions where other systems fail. We apply the time-delay rate equation model in [17,18] of OFB in SLs to investigate the influence of OFB and the ratio τ_r/τ_nr on the dynamics (operation states and route to chaos) and intensity noise level. Intensive computer simulations are employed to investigate the dynamics states and noise of 1550-nm InGaAsP/InP SL with a short external cavity and injection current.

The time-delay rate equations of OFB in SLs are shown in the following section. In 3, we present the procedures of the numerical solution of rate equations. The bifurcation diagrams with OFB strength at several values of the ratio τ_r/τ_nr, the injection current, and the RIN at low-frequency regions are investigated in Section 4. The simulation dynamics, such as time variations with and without noise sources, phase portraits, and RIN under route to chaos operation state, are emphasized. The conclusion is drawn in Section 5.

2. Theoretical Model

The rate equations of the photon number S(t), the optical phase θ(t), and the carrier number N(t), which describe the laser diode with an eternal cavity, are formulated as [17,18]. The used time-delay model is shown in Figure 1.

$$\frac{dS}{dt}=\left\{G-BS-G_{th0}+\frac{c}{n_{r1}L}\ln \left|T\right|\right\}S+\frac{a\xi }{V}N+F_S\left(t\right)$$

(1)

$$\frac{d\theta }{dt}=\frac{\alpha a\xi }{2V}(N-\overline{N})-\frac{c}{2n_{r}L}(\varphi -\overline{\varphi })+F_{\theta }(t)$$

(2)

$$\frac{dN}{dt}=-GS-N\left(\frac{1}{{\tau }_e}\right)+\frac{I}{e}+F_N(t)$$

(3)

$$G=\frac{a\xi }{V}(N-N_g)$$

(4)

where G is the linear gain coefficient, a and N_g are material constants, and ξ is the confinement factor of the optical field into the active region with volume V. α is the line-width enhancement factor, I is the injection current, and $\overline{N}$ is the time-averaged carrier number. The loss coefficient k of the laser diode and mirror loss is used to determine the threshold gain level G_th0:

$$G_{th0}=\left(c/n_r\right)\left\{k+\left(1/2L\right)\ln \left(1/R_f{R}_b\right)\right\}$$

(5)

Based on the density-matrix theory, different models have been advanced to account for gain nonlinearities [26,27]. The coefficient B is the gain saturation parameter.

The carrier lifetime τ_e, which should account for the radiative and non-radiative recombination through radiative decay, non-radiative contribution, and a decay coefficient due to Auger processes, is given by [26]:

$$\frac{1}{{\tau }_e}=B_r(N/V)+\left[A_{nr}+C_{AUG}{(N/V)}^2\right]=\frac{1}{{\tau }_r}+\frac{1}{{\tau }_{nr}}$$

(6)

where A_nr represents the non-radiative contribution emerging from crystal impurities, B_r is the radiative decay parameter, and C_AUG is a decay parameter due to Auger processes [26]. Here, the recombination of the carriers in the rate Equations (1)–(3) is included in terms of the radiative lifetime τ_r. Moreover, non-radiative recombination of the carriers, such as non-radiative contribution emerging from crystal impurities and Auger recombination, is expressed by non-radiative lifetime τ_nr [26,28].

The threshold conditions of the time-delay laser diode are given by [18]:

$$T=1-{\displaystyle \sum _{m=1}{\left(K_{ex}\right)}^m{\left(R_f/\left(1-R_f\right)\right)}^{m-1}}\exp \left\{-jm\psi \right\}\left(S(t-m\tau )/S(t)\right)=\left|T\right|\exp \left\{-j\varphi \right\}$$

(7)

where m is the number of round trips, ψ = φ_ex + φ_f + ωτ is combining all phase variations due to reflection by the external mirror, where φ_ex is the phase retarded from the external mirror, φ_f is the phase from the front facet, ω is the circular frequency emission, and $\tau =2\pi n_{ex}{L}_{ex}/c$ is the round-trip time. R_ex, n_ex, and L_ex are the reflectivity of the external mirror, refractive index, and the length of the external cavity, respectively. The OFB strength K_ex, which is described by the ratio of the reflectivity of the external cavity R_ex to the reflectivity of the front facet R_f as,

$$K_{ex}=(1-R_f)\sqrt{\eta R_{ex}/R_f}$$

(8)

where η is the coupling coefficient of the injected light to the laser cavity. The pretext φ of the real and imaginary parts OFB function T is clearly given by:

$$\varphi =-{\tan }^{-1}\left\{\mathrm{Im}\left[T\right]/\mathrm{Re}\left[T\right]\right\}+\ell \pi$$

(9)

where $\ell$ is an integer.

The solitary laser (laser without OFB) is assumed to operate under CW operation. By taking the OFB terms (K_ex = 0) in Equations (1)–(3), equating the right-hand sides with zero, and then solving the algebraic equations, the characteristics of the steady-state values of the photon number and carrier number are obtained:

$$B{\overline{S}}_{}^3+\left(\left[BV/\left(a\xi \left(\left(1/{\tau }_r\right)+\left(1/{\tau }_{nr}\right)\right)\right)\right]+G_{th0}\right){\overline{S}}_{}^2-\left(\left(I/e\right)-\left(N_{th0}/{\tau }_e\right)+\left(a\xi /V\right)N_g\right){\overline{S}}_{}-\left(I/e\right)=0$$

(10)

While $\overline{N}$ is given by:

$$\overline{N}=\left[\left(\left(a\xi /V\right)N_g\overline{S}+I/e\right)/\left(\left(a\xi /V\right)\overline{S}+\left(\left(1/{\tau }_r\right)+\left(1/{\tau }_{nr}\right)\right)\right)\right]$$

(11)

These values of the photon number and carrier number are used to determine the relaxation frequency f_r of the solitary laser as

$$f_r\approx \left(1/2\pi \right)\sqrt{\left(a\xi /V\right)\left[\left(a\xi /V\right)\left(\overline{N}-N_g\right)+B\overline{S}\right]\overline{S}}$$

(12)

The values of these parameters at injection current I = 2.0I_th0 are $\overline{S}=5\times {10}^5$, $\overline{N}=3.93\times {10}^8$, and f_r = 3.5 GHz.

The functions F_S(t), F_N(t), and F_θ(t) are Langevin noise sources, and $\langle F_S(t)\rangle =\langle F_N(t)\rangle =\langle F_{\theta }(t)\rangle =0$, with the assumption of having Gaussian probability distributions for their correlation functions:

$$\langle F_x(t)F_y(t^{\prime })\rangle =V_{xy}\delta (t-t^{\prime })\hspace{1em}\hspace{1em}x,y=S,N\mathrm{or}\theta ,$$

(13)

Here V_xy are the correlation variances. It can be determined by the mean values of S and N at each carrier transition of Equations (1)–(3) [26]. By using computer random-number generations, the method of simulating these noise sources is given in reference [26].

3. Numerical Calculations Process

In these simulations, the typical values of parameters of InGaAsP/InP lasers emitting at 1550 nm are listed in

Table 1. InGaAsP/InP laser (emitting at 1550-nm) parameters which were employed in the calculations.

Parameter	Value	Unit
External cavity refractive index nex	1.0	---
Gain tangential coefficient a	7.9 × 10−12	m3s−1
Carrier number at transparency Ng	1.33 × 108	---
Active region refractive index nD	3.513	---
Active region length LD	300	μm
Active region volume V	150	μm3
Field confinement factor ξ	0.2	---
Front-facet reflectivity Rf	0.3	--
Back-facet reflectivity Rb	0.8	--

. The threshold current of the solitary SL, I_th0 = 10.8 mA. The rate Equations (1)–(3) are solved numerically by means of the fourth-order Runge–Kutta method. The time step used in the integration is set as ∆t = 5 ps and is above the relaxation frequency f_r = 3.5 GHz and the frequency of the external cavity f_ex = 2.5 GHz at I = 2.0I_th0, to undertake the induced dynamics due to the fine resolution of the OFB. The operation state of the laser is examined after carrying the integration over a period of 6 μs (it is long enough to stabilize the operation of the laser). The integration is first made by ignoring OFB (K_ex = 0.0) (solitary laser) starting from t = 0 until the round-trip time τ. The values of S(t = 0 → τ) and θ (t = 0 → τ) are then used as time-delayed values S(t –τ) and θ (t –τ) for further integration of the rate equations, including OFB terms. Three round trips (m = 3) are considered in these calculations to account for the case of strong OFB. The phase difference ψ is arbitrary and set to be ψ = ωτ, the external cavity length is L_ex = 0.06 m, and the averaged values are $\overline{\varphi }$ = 0 in the present simulation. The RIN spectra are computed directly from the obtained values of the fluctuations of the photon number δS(t) = S(t)-$\overline{S}$ by using the fast Fourier transform as:

$$RIN=\frac{1}{{\overline{S}}^2}\left\{\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}\left[{\displaystyle \underset{0}{\overset{\infty }{\int }}\delta S(t)\delta S(t+\tau )e^{j\omega \tau }d\tau }\right]dt}\right\}=\frac{1}{{\overline{S}}^2}\left\{\frac{1}{T}{\left|{\displaystyle \underset{0}{\overset{T}{\int }}\delta S(\tau )e^{-j\omega \tau }d\tau }\right|}^2\right\}$$

(14)

where $\overline{S}$ is related to the dc-value of S(t). $\overline{S}$ is time-averaged based on the photon number and has been calculated numerically. The RIN calculations were executed when the steady state (CW operation) is gained (t ≈ 4~6 μs).

4. Simulation Results Analysis, and Discussions

4.1. Bifurcation Diagrams with OFB Strength

Investigation of the influence of the radiative and non-radiative recombination lifetime ratio τ_r/τ_nr on the operation states of the InGaAsP/InP laser is conducted by calculating the bifurcation diagram of the photon numbers S(t) with the strength of OFB K_ex and injection current (I-I_thc). The bifurcation diagrams—with neglecting noise sources in the Equations (1)–(3), and at four values of τ_r/τ_nr = 2.0, 0.2, 0.02, and 0.002, and two values of the injection current, (I-I_thc) = 1.0 mA (close to the threshold due to the strong effect of spontaneous emission, which enhance influence of the fluctuations), and 43.0 mA (well above the threshold, because the coherency of the laser radiation increases and the fluctuations effect is lessened)—are depicted in Figure 2a–d, and Figure 3a–d, respectively. In these figures, we will explore the SL operation states over a wide range of the ratio τ_r/τ_nr, OFB strength, and injection current.

Figure 2 and Figure 3 show that under weak OFB, the solution of the rate equations is stationary, and the laser operates in CW state, which means that the laser is stable at all values of the τ_r/τ_nr ratio. By enhancing the strength of the OFB, the solution bifurcates first into a balanced PO, which is characterized by undamped periodic oscillation with a frequency f_po that is not necessarily equal to the frequency of the relaxation oscillation f_r of the laser without OFB [29]. The onset point of the bifurcation is called an HB point and is identified by a frequency f_po(H) that is not equal to the relaxation oscillation frequency f_r [29]. The point of HB, which is indicated in Figure 2a–d and Figure 3a–d by arrows, is shifted into the above values of the OFB K_ex by increasing the value of the ratio τ_r/τ_nr; it increases from 0.0018 to 0.0072 by increasing the ratio τ_r/τ_nr from 0.002 to 2.0, respectively, as shown in Figure 2a–d. At a higher injection current, when (I-I_thc) = 43.0 mA, it increases from 0.014 to 0.034 by increasing the ratio τ_r/τ_nr from 0.002 to 2.0, respectively, as shown in Figure 3a–d. Further increasing the strength of the OFB, the results in Figure 2 and Figure 3 show that the solution of the Equations (1)–(3) bifurcates into many bifurcation points, which is followed by a chaotic operation, and the type of these bifurcations depend on the ratio τ_r/τ_nr and injection current values.

In Figure 2a, when (I-I_thc) = 1.0 mA, and the ratio τ_r/τ_nr = 2.0, PO region is followed by chaotic state, which means that the route to chaos is PO. Further decrease in the ratio τ_r/τ_nr to be = 0.2, 0.02, and 0.002 PO is followed by PD then chaos, as indicated in Figure 2b–d.

By increasing the injection current, when (I-I_thc)= 43.0 mA, and when the ratio τ_r/τ_nr = 2.0, multiple bifurcation points are appeared; three bifurcation points indicating to unstable transition to chaos operation. Such type of route to chaos is called 3SH route to chaos, as indicated in Refs. [30,31]. At lower values of the ratio τ_r/τ_nr = 0.2, 0.02, and 0.002, Figure 3b–d show two routes to chaos. The first route to chaos is PO, which is followed by multiple bifurcation and the state is moved to unstable operation or chaos operation [32]. By increasing the OFB strength K_ex, the coherence collapse or first chaos region is followed by the second route to chaos, which is appeared as three bifurcations indicating to second unstable transmission to chaos. Such a second transmission to chaos is called 3SH route to chaos [30,31]. That means, the route to chaos of SLs varies from PO, PD to SH route to chaos relying on the ratio τ_r/τ_nr and the injection current.

By increasing the strength of the OFB, K_ex, the operation state is pulled to transition to chaos. The range of the chaos regime is decreased and is interspersed with stable regions by varying the ratio τ_r/τ_nr, as shown in Figure 1, but by increasing the injection current from (I-I_thc) = 1.0 mA to (I-I_thc) = 43.0 mA, the width of the chaotic region becomes narrower and the density of the bifurcation points in the chaos region becomes higher, as shown in Figure 3. This means that increasing the injection current in the rate equations will lead to narrowing the chaotic region and increasing the width of the stable regions, which lead to more stability in the laser.

By enhancing the OFB beyond the chaotic region, we found that the laser operates in the CW region (K_ex > 0.9) when (I-I_thc) = 1.0 mA, as shown in Figure 1. At larger values of the injection current and when (I-I_thc) = 43.0 mA, and under strong OFB, a narrow region of chaos at K_ex = 1.4) appears, and the laser oscillates in CW or PO depending on OFB strength. Under strong feedback and by raising the ratio τ_r/τ_nr and injection current (I-I_thc), the states of the laser become more stable and attracted toward to CW or PO operation, as shown in Figure 2a.

From Figure 2 and Figure 3 we have found that the impacts of the OFB and the ratio τ_r/τ_nr on the states of the laser are opposite. For instance, we have shown that the range of the CW and PO regime with weak OFB and chaos regime with moderate OFB becomes tighter by increasing the ratio τ_r/τ_nr, which is stabilizing to the laser operation state. In addition, the transition to chaos is modulated to stable route to chaos from SH to PD or PO route to chaos. By enhancing the strength of the OFB, the states move to a more stable regime, and by increasing the ratio τ_r/τ_nr, the chaos region modulated to CW and PO state. A further increase in OFB strength only causes the laser states to be more complicated and unstable. In contrast, by increasing the ratio τ_r/τ_nr, it reverses the states of the laser and causes the route to chaos to be more stable, and the laser states regain their constancy, particularly when the laser operated in the strong OFB regime.

Employing bifurcation diagrams represented in Figure 2 and Figure 3, the feedback level at HB point is calculated as a function of the ratio τ_r/τ_nr and is plotted in Figure 4, at two values of injection current (I-I_thc) = 0.1 mA and = 43.0 mA. As shown in Figure 4, the variations of the OFB strength at HB point are almost linear functions with positive slope of τ_r/τ_nr, which means when increasing the OFB level, one needs to increase the value of the ratio τ_r/τ_nr required to stabilize a PO, PD, SH or a chaos state. Near threshold, the variation of feedback level of the HB point with τ_r/τ_nr is very slight and almost constant at lower values of τ_r/τ_nr. By increasing the injection current (I-I_thc) = 43.0 mA to be substantially far from the threshold, the variation is not changed but the feedback level becomes higher by almost one order of measure, as shown in Figure 4, which shows what was represented in Figure 2 and Figure 3, that the feedback level at HB point increases by increasing the injection current at constant value of the ratio τ_r/τ_nr. Figure 2, Figure 3 and Figure 4 demonstrate that increasing the ratio τ_r/τ_nr and OFB strength returns the laser to a steady state operation.

By increasing the ratio τ_r/τ_nr, the OFB strength at HB point and route to chaos increases, as shown in Figure 3. This is because by increasing the value of the ratio τ_r/τ_nr, the threshold current of the laser is increased according to the equation of the laser threshold current in terms of τ_r and τ_nr, which consider the effect of the external OFB [33],

$$I_{thc}=e\left(1/{\tau }_r+1/{\tau }_{nr}\right)\left\{N_g+(V/a\xi )\left[G_{th0}-\left(c/n_{r}L\right)\ln (T)\right]\right\}$$

(15)

From the results shown in Figure 2, Figure 3 and Figure 4, we think that operating the SL in the strong OFB regime, higher values of the ratio τ_r/τ_nr, and injection current far from the threshold contributes to designing a stable and noiseless laser diode with high performance.

4.2. Time Variation, Phase Portrait, and RIN of S(t)/$\overline{S}$ at Route to Chaos

The time variations, phase portrait, and RIN of the emitted photons number S(t)/$\overline{S}$ from the back facet of the SL for various operation states that characterize the routes to chaos are shown in Figure 2 and Figure 3.

Figure 5 simulated the variation of the photon number S(t)/$\overline{S}$ with time with and without noise sources and with and without OFB (solitary laser (dotted-red curve)), phase portrait, and RIN spectrum when (I-I_thc) = 1.0 mA. The laser exhibits oscillations with single peak (PO route to chaos) when the ratio τ_r/τ_nr = 2.0 and K_ex = 0.016 (Figure 5(a1)), and by decreasing the ratio τ_r/τ_nr to τ_r/τ_nr = 0.2, 0.02, and 0.002 and K_ex = 0.01 (Figure 5(a2–a4)), the single peak oscillations change to oscillations with two peaks (PD route to chaos). The oscillations shown in Figure 5(a1–a4) are confirmed by the limit cycle (LC) and PD attractors that characterize the phase portrait of Figure 5(b1–b4). The corresponding time variation of S(t)/$\overline{S}$ with noise and the RIN spectrum of oscillations at the route to chaos point is plotted in Figure 5(c1–c4,d1–d4), respectively. The slight phase shift appeared in Figure 5(c1–c4) by considering the ratio τ_r/τ_nr interpreted to a slight transfer in the frequency in the spectra of the RIN in Figure 5(d1–d4). The spectrum is indicated by the frequency of the periodic oscillation f_po = 1.8, 1.39, 1.28, and 1.26 GHz, which is smaller than relaxation oscillation frequency f_r = 2.3, 1.46, 1.36, and 1.29 GHz, when τ_r/τ_nr = 2.0, 0.2, 0.02, and 0.002, respectively. The OFB regulates the frequency of the relaxation oscillation f_r of the laser to the frequency of PO f_po, as mentioned in the reference [31], which is increased by increasing the ratio τ_r/τ_nr. Moreover, the RIN at low frequencies is higher (almost one order of measure) than that of the noise of the solitary laser (red dashed curve) through the ratio τ_r/τ_nr.

Under higher values of (I-I_thc) = 43 mA, and when it becomes well above the threshold, the first route to coherence collapse state of the SL, when τ_r/τ_nr = 2.0 and K_ex = 0.0068, is characterized by two different peaks in the time variation of the S(t)/$\overline{S}$, as presented in Figure 6(a1). When τ_r/τ_nr = 0.2, 0.02, and 0.002 and K_ex = 0.0022, respectively, the transition to chaos change to PO route to chaos with one peak. The PD and LC attractors representing the PD and PO operation are shown in the phase portrait in Figure 6(b1–b4), respectively. Figure 6(c1–c4,d1–d4) show the time change of the S(t)/$\overline{S}$ with noise, and the RIN power spectra with peaks around the PO frequency f_po and its harmonics, characterizing the PD and PO dynamics. The slight phase shift appearing in Figure 6(c1–c4) is reduced by increasing the injection current. The spectrum is identified by the frequency of the periodic oscillation f_po = 6.8, 6.4, 6.25, and 6.2 GHz, which is smaller than relaxation oscillation frequency f_r = 6.9, 6.7, 6.49, and 6.4 GHz, when τ_r/τ_nr = 2.0, 0.2, 0.02, and 0.002, respectively. The RIN at low frequency has increased the level of the solitary laser by more than half an order of measure. In the route to chaos regimes, the ratio τ_r/τ_nr causes a frequency transfer-induced phase transfer with respect to the frequency in the state of the solitary laser.

These telling results are, to the authors’ knowledge, a new achievement in the dependence of the lasing frequency transfer on the OFB strength and the ratio τ_r/τ_nr of the SL.

Finally, we can conclude that the increase in the ratio τ_r/τ_nr under injection current far from threshold converts the complicated and unstable system to a stable state with periodic oscillation and low RIN.

Figure 7 represents the second transfer to chaos, when increasing the OFB strength K_ex to be K_ex = 0.04, 0.03, and 0.028, and τ_r/τ_nr = 0.2, 0.02, and 0.002, respectively, the time change of the S(t)/$\overline{S}$ (without noise) characterized by multiple different peaks (three peaks). The multiple different peaks shown in Figure 7(a1–a3) represent the 3SH route to chaos. Figure 7(a1) (τ_r/τ_nr = 0.2 and K_ex = 0.03) shows the state that the complexity of the system decreases the irregularities in S(t)/$\overline{S}$ took the laser oscillations near to PD transfer to chaos, which is characterized by the two peaks approximation. The RIN spectra show multiple, three, two, and one peaks, that correspond to multiple, three, two, and one various peaks of S(t)/$\overline{S}$ appearing in Figure 7(d1–d3), respectively. In Figure 7(b1–b3), tours, which represent the multiple SH, PD (approximation), and 3SH attractor’s route to chaos, appear in the phase portrait, which confirm the result shown in Figure 7(a1–a3). The corresponding time change of the S(t)/$\overline{S}$ with noise, and the RIN power spectra with peaks around the frequency of the PO f_po, and its harmonics characterizing the PD and SH dynamics, are shown in Figure 7(c1–c3,d1–d4). The slight phase shift in this case of second route to chaos is decreased and approximately disappeared. The spectrum is characterized by the frequency of the periodic oscillation f_po = 6.4, 6.25, and 6.2 GHz, which is smaller than relaxation oscillation frequency f_r = 6.7, 6.49, and 6.4 GHz, when τ_r/τ_nr = 0.2, 0.02, and 0.002, respectively. The low-frequency RIN behaves like the first route to chaos and is increased approximately by one order of measure more than the level of the solitary laser.

The RIN increases in the low-frequency region by decreasing the ratio τ_r/τ_nr, as shown in Figure 5, Figure 6 and Figure 7. Moreover, the slight phase transfer-induced frequency transfer with respect to the solitary laser frequency is decreased. This impact of the ratio τ_r/τ_nr on the RIN is more efficient when injection current is near to the threshold I_th because of the fluctuations enhancement. When the current is selected to be far from the threshold, the effect of the ratio τ_r/τ_nr is decreased due to increasing the coherency of the laser radiation and weakness of the fluctuations.

4.3. Influence of OFB and the Ratio τ_r/τ_nr on the Low-Frequency RIN

The variations of the average of the RIN at low frequencies of f = 10 MHz, with a change in the strength of the OFB corresponding to results shown in Figure 2 and Figure 3, is plotted in Figure 8. We can see under the weak feedback region when the injection current (I-I_thc) = 1.0 mA, that the average RIN is enhanced to more than three orders of measure than that of the quantum noise level, which resembles the solitary laser (dotted black line). By increasing the injection current to (I-I_thc) = 43.0 mA, the average RIN is reduced to the quantum noise level in all cases. When τ_r/τ_nr = 2.0, the low-frequency RIN (black line) is the lowest value and suppressed approximately to the solitary laser noise level. By decreasing the ratio τ_r/τ_nr = 0.2, 0.02, and 0.002, the low-frequency noise becomes higher (more than half an order of measure). In the regime from HB point to the route to chaos, the RIN has diminished approximately to the level of the solitary laser (less than half an order of measure), as displayed in Figure 5, Figure 6 and Figure 7. Under moderate OFB, the disturbance characterizing the chaotic operation raises the RIN to higher levels above the quantum level (almost six orders of measure) in both cases of injection current. As displayed in Figure 8, the RIN in strong OFB is repressed approximately to or lower than the quantum noise (dotted black line), depending on the value of the ratio τ_r/τ_nr. By increasing the ratio τ_r/τ_nr and the injection current, the chaos operation varied to CW operation, as displayed in Figure 3a, and the RIN is suppressed close to the quantum noise.

Figure 9 shows the frequency spectra of the RIN under strong OFB when K_ex = 2.0. As found in the figure, the RIN spectra display that the RIN at low frequency of the CW operation is suppressed by enhancing the ratio τ_r/τ_nr and injection current to be near to or lower than the quantum noise level. In the higher frequency region, peaks can be identified at frequencies near to the frequency of the external cavity f_ex = 2.5 GHz, relaxation frequency f_r = 3.5 GHz, and their harmonics. Considering and increasing the ratio τ_r/τ_nr parameter in the Equations (1)–(3) and adjusting the semiconductor laser to operate with strong regime and higher injection current drives the laser to change its state from instability (chaotic) to stable (CW), which oscillates with the frequency of external cavity f_ex and frequency of relaxation f_r. This influence of the ratio τ_r/τ_nr on RIN is more efficient when injection current is near to the threshold value because of the fluctuations enhancement. By increasing the current (far from threshold), the ratio τ_r/τ_nr is decreased due to increasing the laser coherency and weakness of the fluctuations [24].

5. Conclusions

We studied the effect of the OFB strength and the ratio of the radiative and the non-radiative recombination lifetime, τ_r/τ_nr, on the state, transfer to chaos, and relative intensity noise RIN, of long wavelength semiconductor lasers. Our calculations were employed to a 1550-nm InGaAsP/InP laser at several values of the current near and far enough from the threshold value. The operation state with OFB, route to chaos, and the laser operations are categorized by the bifurcation diagrams, time-variations, phase portrait, and the RIN power spectrum as functions of the ratio τ_r/τ_nr and injection current. The simulation results showed that considering the variations of the ratio τ_r/τ_nr causes important impacts on the behavior of the states and the route to chaos of the lasers. We found that the OFB strength (when the transition from CW to PO occurs) rises with the raise in the ratio τ_r/τ_nr. Varying the ratio τ_r/τ_nr was found to change the route to chaos of the laser from SH or PD type to PD or PO. The route to chaos state of SLs is identified in three distinct operating regions, namely, PO, SH, and PD, depending on the value of the ratio τ_r/τ_nr and the injection current. Under higher injection current, and by raising the ratio τ_r/τ_nr, the laser route to chaos changes from SH to PO route to chaos. In the transfer to chaos regime, the OFB strength and the ratio τ_r/τ_nr causes a slight shift in the frequency-induced phase shift with respect to the solitary laser frequency. The obtained results, as far as we know, are new studies on the dependence of the lasing frequency transfer on the ratio τ_r/τ_nr and the OFB. By operating the laser in the CW or PO regime (weak OFB) the average value of the RIN is reduced to the quantum noise, which is realized when the ratio τ_r/τ_nr increases. Entering a strong OFB regime, at larger values of the ratio τ_r/τ_nr, and injection current far from the threshold, the laser state is stabilized and is changed from a chaos state to a stable state (CW state). In addition, The RIN is suppressed near to or less than the quantum noise level. The complexity of the states of the SLs can be decreased by considering the influence of OFB strength, the ratio τ_r/τ_nr, and injection current, which supports designing a high-performance SL. A good understanding of the impact of the carrier recombination mechanisms in SLs on the dynamics and intensity noise is essential for designing SLs with high static and dynamic performance. In this paper, we determined the best ranges of the ratio t_r/t_nr, OFB strength, and injection current, which can be used to stabilize the semiconductor laser and operate it with low noise, improving the performance of SLs.