Analysis of SFM dynamics in solitary and optically-injected VCSELs

A. Homayounfar; M. J. Adams

doi:10.1364/OE.15.010504

1. Introduction

Polarisation properties of vertical-cavity surface-emitting lasers (VCSELs) offer potential applications in telecommunication such as polarisation switching (PS) and polarisation bistability. The fundamental transverse mode of a VCSEL can possess two orthogonal polarisations with corresponding effective refractive indices (propagation constants) whose difference in magnitudes can be defined by a birefringence rate. PS between the two linearly polarised (LP) modes can be achieved either by changing the pumping current (see [1] and references therein) or introducing polarised optical injection [2, 3], both of which can be described by the spin-flip model (SFM) [4, 5]. Optical injection in both edge-emitting lasers and VCSELs has important communications applications; recent results for injection-locked VCSELs have demonstrated enhanced frequency response and improved performance in both analogue and digital communications [5,6], and for the first time a double resonance frequency response was observed when polarisation effects were investigated [7].

In recent publications [8, 9] we have used the SFM to analyse the behaviour of a slave VCSEL subject to polarised optical injection. We found a standard range of values for birefringence, spin relaxation rate and pumping for which the stability map of detuning versus injection strength contains a rich variety of polarisation-sensitive behaviour, including injection locking, limit cycle, chaos and quasi-stability for elliptically-polarised (EP) and LP output. Outside that range, EP regions are replaced by LP regions. In the present paper we focus our investigation on the standard range of parameters that imply both EP and LP characteristic behaviour in the stability map. For optical injection in slave VCSELs, it is important to know if the solitary laser is in linear stability (LS), elliptical stability (ES) or instability. The laser variables should be considered as the initial values when external polarised light is injected into the slave laser. Therefore, in this paper we first use the SFM to analyse solitary VCSELs using the same range of parameters.

In section 2 the fundamental SFM equations are discussed and relationships are given between normalised variables in the SFM and the conventional rate equations when polarisation is neglected. Section 3 deals with the parameter values that we use for the key variables. The stationary solutions of the equations and effects of ES are discussed in section 4. To the best of our knowledge published investigations of ES using the SFM are limited to two specific cases: one where the spin relaxation rate is 50 ns^-1 [5], in which case ES is found in a very narrow domain close to the LS region, and the other where high values of birefringence and spin relaxation rates and pumping are assumed [10]. In contrast the influence of intermediate-range values of pumping, spin relaxation and birefringence rates on ES is analysed in this paper.

The other novel feature of the present contribution is the investigation of the relaxation oscillation frequency (ROF) for ES as compared to that for LS; numerical results are given in section 5. The ROF is related to the maximum modulation bandwidth and it is well-known that optical injection locking can increase the ROF and hence the bandwidth (see [11, 12] and references therein). However the effects of polarised optical injection on the ROF have not been studied. Indeed only a few investigations on the effects of polarised injection have been made since the first experimental report of stable injection locking within a large frequency detuning range (80 GHz) when the injection light is polarised parallel to the polarization of the solitary VCSEL [13]. In section 6 we discuss our results for ES in the context of recent theoretical and experimental publications on polarised optical injection in VCSELs. To investigate details of EP behaviour, in section 7 we analyse y-polarised injection whilst the solitary laser is either linearly x-polarised (orthogonal injection) or in ES. Taking account of the conditions of the solitary VCSEL in LS and ES we investigate different types of EP and LP states in the stability map for polarised optical injection. In section 7 we also discuss the influence of fundamental parameters of birefringence, pumping and spin relaxation rate on the stability map for optical injection when the solitary laser is either in LS or ES.

2. Theory

The SFM equations in terms of orthogonal linear field amplitudes and phases can be derived from the rate equations for complex circularly-polarised (CP) fields [4, 5, 14] to give the forms:

\frac{{dE}_{y}}{dt} = \frac{1}{2 τ_{p}} [(N - 1) E_{y} + m E_{x} \frac{\sin (ε - Δ ϕ)}{\cos ε}] + γ_{a} E_{y}

\frac{{dE}_{x}}{dt} = \frac{1}{2 τ_{p}} [(N - 1) E_{x} - m E_{y} \frac{\sin (ε + Δ ϕ)}{\cos ε}] - γ_{a} E_{x}

\frac{d Δ ϕ}{dt} = - \frac{m}{2 τ_{p} \cos ε} [\frac{E_{x}}{E_{y}} \cos (ε - Δ ϕ) + \frac{E_{y}}{E_{x}} \cos (ε + Δ ϕ)] + 2 γ_{p}

\frac{dN}{dt} = - γ [N (1 + {E_{x}}^{2} + {E_{y}}^{2}) - (η_{+} + η_{-}) - 2 m E_{y} E_{x} \sin Δ ϕ]

\frac{dm}{dt} = - γ_{s} m + γ (η_{+} - η_{-}) - γ [m ({E_{x}}^{2} + {E_{y}}^{2})] + 2 γ N E_{y} E_{x} \sin Δ ϕ

where ε=tan^-1(α) and Δϕ=2(γ_p-αγ_a)t+ϕ_y-ϕ_x. In these equations E_x and E_y are orthogonally polarised field amplitudes and ϕ_x, ϕ_y are the corresponding phases. N and m are, respectively, the total carrier concentration and difference in concentration between carriers with spin-up and spin-down. The decay rates due to electron recombination, gain anisotropy, birefringence and spin relaxation are denoted by γ, γ_a, γ_p and γ_s, respectively, τ_p is the photon life time and α is the linewidth enhancement factor (Henry factor). η₊ and η_- are CP pumping terms; when these are equal (and defining 2η=η₊ + η_-) the SFM reproduces the situation of electrical pumping [15]. The SFM uses normalised equations with dimensionless variables defined such that the values of E_x, E_y and N are close to unity. For the case of electrical pumping, in the limit of vanishing polarisation effects equations (1)–(5) should reduce to two conventional rate equations [16]. To achieve this by decreasing gain anisotropy and birefringence causes the VCSEL to experience LP stability for E_y (E_x) whilst m and E_x (E_y) are both zero.

3. Values for key parameters

The gain anisotropy rate γ_a indicates the difference of the two gain constants for E_x and E_y, and the birefringence rate γ_p is half of the difference between the resonance angular frequencies of the two polarized modes (ω_y and ω_x), i.e. γ_p=(ω_y-ω_x)/2. When high values for the spin relaxation and birefringence rates are assumed, the SFM equations can be reduced to a simpler set of equations where self- and cross-saturation parameters are used [1, 14]; equations (1)–(5) can also be reduced to two equations by using asymptotic methods as γ_a, γ_p→0 whilst γ_s is on the order of a few hundred ns^-1 [17]. Neither of these sets of conditions is satisfied in the present paper where the birefringence rate does not exceed 50 ns^-1 or fall below 9 ns^-1. This relatively modest range for γ_p satisfies the condition for using equal gain constants of the two orthogonal polarised fields, which justifies our assumption that γ_a=0.

The spin relaxation rate γ_s depends on the material used for the active region of the VCSEL, as well as the electron confinement energy in the quantum wells [18, 19] and the temperature [20, 21]. Experimentally measured values for room temperature have been reported in the range between 6 ns^-1 and 55 ns^-1 for GaAs/AlGaAs quantum wells [18–22] and in the range 50 ns^-1-280 ns^-1 for InGaAs/InP quantum wells [18, 19, 21, 23]. Results for (110)-oriented quantum wells show even slower spin relaxation rates down to around 1 ns^-1, which is of the order predicted to be required for efficient threshold reduction in a room-temperature spin-VCSEL [24]. In this paper we use values for the spin relaxation rate of 20 ns^-1 and 40 ns^-1, which are in the middle of the standard range of measured values for GaAs/AlGaAs quantum wells. Further justification for this choice of parameter values will be given in the next section.

4. Elliptically and linearly polarised stability

Figures 1 and 2 show the regions of stability on plots of normalised pumping versus birefringence rate for a VCSEL with the parameters of Table 1 and γ_a=0. The LS in Figs. 1 and 2 is shown for x(y) polarised fields as LS_x (LS_y) which is denoted in dark (light) grey area. The LS_x (E_x=E_st and E_y=0) and LS_y (E_y=E_st and E_x=0) are the two linearly polarized steady-state conditions for which one field component is zero and the other takes the value E_st=(2η-1)^1/2; these solutions also have N=1 and m=0 [5]. For fixed γ_s and 2η there is a minimum (maximum) value for birefringence γ_px-min (γ_py-max) [5], which defines the starting (ending) point of the LS_x (LS_y) region.

γ_{px - min} = \frac{α}{2 τ_{p} (1 + \frac{γ_{s}}{γ (2 η - 1)})}

γ_{py - max} = \frac{[γ_{s} + γ (2 η - 1)]}{2 α}

Table 1. Parameter values for two VCSELs with different spin relaxation rates.

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Fig. 1. Stability diagram for a VCSEL with parameter values from Table 1 and spin relaxation rate γ_s=20 ns^-1. The four regions denote linearly polarised stability for x-LP (LS_x, dark grey) and y-LP (LS_y, light grey), ES (orange) and instability (white).

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Fig. 2. Stability diagram as in Fig. 1, but for a larger value of spin relaxation, γ_s=40 ns^-1, which shifts the critical point where the system is either at LS_x or LS_y towards higher values of pumping. The red area below the critical point now corresponds to linear bistability (LB). According to the numerical results, below the critical point the extent of LS_y slightly exceeds the boundary of (6)

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We have chosen a range of normalised pumping and birefringence such that the separation between boundaries of the regions for LS_x and LS_y does not exceed 5 GHz (10π rad/ns). By comparing the equations (1)–(4) with conventional rate equations [16] the expression for the normalised pumping term 2η can be rewritten in terms of the ratio of pumping J to threshold J_th

2 η = 1 + (\frac{J}{J_{th}} - 1) (1 + τ_{p} n_{o} G_{n})

where n_o and G_n are the transparency carrier density and gain coefficient, respectively. It is possible to see 2η is always larger than unity for currents above threshold. Applying the condition γ_px-min - γ_py-max≤10π on the boundaries between LS_x and LS_y, for the range of (J/J_th - 1) between 30% (2η=2.026) and 75% (2η=3.565), the corresponding range of spin relaxation rate, γ_s, is found from equations (6) and (7) to lie between 21 ns^-1 and 46 ns^-1, respectively, for the values of Table 1. The most sensitive area of Figs. 1 and 2 for any small changes of the parameters is the critical point where γ_px-min=γ_py-max. At the critical point a pumping range between, for example, 2η=1.1 and 2η=1.4 (3% and 11% above threshold) corresponds to a γ_s range between 21 ns^-1 and 42 ns^-1, respectively. These considerations give further justification for our choice of γ_s=20 ns^-1 and γs=40 ns^-1 in Figs. 1 and 2, respectively.

The boundary of the LS_x (LS_y) region is denoted by a solid (dash-dot) line in Figs. 1 and 2. LS_x (LS_y) is found in the region to the right (left) of the solid (dash-dot) curve corresponding to equation 6 (7). The SFM calculations also predict that the system can experience an unstable state, which has been indicated by the white region (above the critical point) between the dash-dot and dashed lines of Figs. 1 and 2. In the unstable region periodic behaviour is found for all the variables E_x, E_y, m, N and Δϕ. In the region where LS_x and LS_y overlap, i.e. where γ_px-min < γ_py-max (between the solid and dash-dot lines and below the critical point), the VCSEL exhibits linear bistability (LB). In the LB region of Fig. 2, denoted by the red area below the critical point, for increasing pumping current the VCSEL switches from the x-polarised mode (LS_x) to the y-polarised mode (LS_y), i.e. type II switching (switching from the lower frequency mode, ω_x, to the higher frequency mode, ω_y)[1,5,25]. However, in Fig. 1, where γ_s=20 ns^-1, the VCSEL is found to exhibit LS_x in the small region where LS_x and LS_y overlap; the reason for LB not occurring here is at present unknown.

In Fig. 1 the region of ES (0 < E_y < E_x < E_st) is shown between dashed and solid lines above the critical point. In Fig. 2 the ES region is reduced in size and closer to the LS_x region compared to Fig. 1. In other words, increasing γ_s has the effect of decreasing the region of ES in size because of the fact that the faster coupling rate (γ_s) between the two carriers with opposite spins causes m to approach zero and reduces the EP effects. Hence, in order to deal with a larger ES region we choose the minimum value for γ_s in the range considered here (γ_s=20 ns^-1) and investigate the magnitudes of the steady-state variables. Fig. 3 shows an enlarged plot of the area of ES of Fig. 1 with points (γ_p, 2η) denoted by letters of the alphabet. These points are used as the key to the polarisation and the variables m and ΔN, respectively in Figs. 4–6 where calculated results are plotted against birefringence rate for fixed values of pumping, 2η=1.4 (light blue), 1.5 (pink), 1.6 (grey), 1.7 (dark blue), 1.8 (green), 1.9 (red), 2.0 (black) and 2.1 (orange). The variable on each of the curves in Figs. 4–6 achieves its maximum at the point of minimum birefringence for the ES region, γ_p-EPmin, and the locus of these points forms the boundary of ES in Fig. 3 (the dashed line).

Fig. 3. Enlarged plot of the region of ES of Fig. 1 with points (γ_p, 2η) denoted by letters of the alphabet. The inset shows the region for normalised pumping below 2η=1.4.

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To measure the strength of ES it is convenient to use the polarisation, S, defined in terms of right (left) circularly polarised fields, E₊ (E_-) as

S = \frac{∣ E_{+}^{2} ∣ - ∣ E_{-}^{2} ∣}{∣ E_{+}^{2} ∣ + ∣ E_{-}^{2} ∣} = - 2 \frac{E_{y} E_{x}}{E_{x}^{2} + E_{y}^{2}} \sin Δ ϕ

where the conventional relations between circularly and linearly polarised fields [5,8] have been used. It is clear that S=0 corresponds to linear polarisation and S=1 (-1) corresponds to right (left) circular polarisation. Figure 4 shows the variation of |S| with birefringence for the pumping rates specified above.

From the steady state condition in (1) and (2) (with γ_a=0) we can derive a relationship between E_y and E_x as

E_{y}^{2} = E_{x}^{2} \frac{\sin (Δ ϕ - ε)}{\sin (ε + Δ ϕ)}

Fig. 4. The polarisation |S| as a function of birefringence rate γ_p for 2η=1.4 (light blue), 1.5 (pink), 1.6 (grey), 1.7 (dark blue), 1.8 (green), 1.9 (red), 2.0 (black) and 2.1 (orange). The dashed line of Fig. 3 is the starting point of the region of ES (γ_p-EPmin, 2η), where each curve has its maximum value for |S|.

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It is clear that as ε tends towards zero, ES approaches CP stability. Equation (10) in ES implies 0<ε≤Δϕ, and it follows that E_y < E_x. The numerical results reveal that in ES Δϕ is never far from ε and hence it is clear that the elliptical strength S is limited. According to Fig. 4 the absolute value of S decreases with increasing birefringence at a fixed pumping rate. S tends to zero at γ_px-min, which lies on the solid lines of Figs. 1, 2 and 3, and is given by Δφ=ε=tan^-1(α).

Unlike LS, in ES the two carrier densities can be written as N=1+ΔN and m, where ΔN and m are of the same order. According to Figs. 5 and 6, at a fixed pumping rate m and ΔN decrease with increasing birefringence because higher γ_p causes more energy transfer from E_y to E_x and hence the system approaches LS. This continues until m and ΔN both approach zero at the boundary of LS_x (solid line in Fig. 3) where the entire energy of E_y transfers to E_x.

Fig. 5. Absolute values of difference between the carriers with spin-up and spin-down, m, as a function of birefringence rate γ_p for the same values of pumping as Fig. 4.

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Fig. 6. Total carrier density N as a function of birefringence rate γ_p for the same values of pumping as Fig. 4.

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The steady-state solution for m in (5) (with η₊=η_-) is given by:

m = \frac{2 \frac{γ}{γ_{s}} {NE}_{y} E_{x} \sin Δ ϕ}{1 + \frac{γ}{γ_{s}} (E_{x}^{2} + E_{y}^{2})}

Equation (11) shows why m approaches zero for linear stability, where E_y=0(E_x=0) for LS_x (LS_y).

It is of interest to estimate the maximum value of m for small values of the coupling between the spin polarised carriers, i.e. the spin relaxation rate (in the case of Figs 3–6, γ_s=20 ns^-1). In ES, noting that γ_s is at least 20 times bigger than γ, it follows from equation (11) that m can be approximated as

m ≅ 2 \frac{γ}{γ_{s}} (1 + Δ N) E_{y} E_{x} \sin Δ ϕ

The maximum value for m can be found when sinΔϕ and E_yE_x have their maximum values, which is achieved by using E_y=E_x=[(2η-1)/2]^1/2 in (12) (in fact, as mentioned above, CP is not possible because ε≠0, so this may give an overestimate of the maximum value of m)

m_{max} < (2 η - 1) (1 + Δ N) \frac{γ}{γ_{s}}

At the boundary between regions of instability and ES (denoted by a dashed line in Fig. 3) m and ΔN have their maximum values. For example point χ for 2η=2.1 has the value m_max ≈0.03 on Fig. 5, and this is within the estimate of (13) which gives m_max <0.05.

5. Analysis of relaxation oscillation frequency in ES

In the regions of LS, since m=0 and one field component is zero, the analysis of relaxation oscillation frequency (ROF) is similar to that for the conventional equations of lasers when polarisation properties are neglected. The small-signal LP ROF, ν _LRO, and the damping rate, γ_d, are given by (Appendix A gives the details of this derivation):

ν_{LRO} = \frac{1}{2 π} \sqrt{\frac{(2 η - 1) γ}{τ_{p}}}

γ_{d} = η γ

The result for ν _LRO in (14) states the frequency of exchanging energy between the total carrier density, N, and one field amplitude (E_y in LS_y or E_x in LS_x), until the laser reaches the steady-state. Note that ν _LRO is independent of γ_p. Figure 7(a) shows the calculated variation of EP ROF, ν _ERO, with birefringence rate for values of pumping, 2η=1.4 (light blue), 1.5 (pink), 1.6 (grey), 1.7 (dark blue), 1.8 (green), 1.9 (red), 2.0 (black) and 2.1 (orange). Increase of birefringence rate causes a faster exchange of energy between E_x and E_y, so that ν _ERO increases whilst E_y and m approach zero at γ_px-min. Figure 7(b) gives the same results in terms of the ratio ρ=ν _ERO(γ_p,2η)/ν _LRO(2η). Again the alphabet points (γ_p, 2η) in Fig. 7 correspond to those in Figs. 3–6. For a fixed value of pumping it is always the case that ν _ERO< ν _LRO. Table 2 contains brief summaries of ROF behaviour in the entire map of Figs. 1 and 2.

Fig. 7. (a) Elliptical relaxation oscillation frequency, ν _ERO, and (b) ratio of elliptical ROF to linear ROF [ρ=ν _ERO(γ_p,2η)/ν _LRO(2η)] as a function of birefringence rate γ_p for the same values of pumping as Fig. 4.

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Table 2. Summary of ROF behaviour for Figs. 1 and 2. For small (large) values of γ_s the area corresponding to γ_px-min < γ_p < γ_py-max is LS_x (LB) in Figs. 1 and 2.

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6. Polarised optical injection in VCSELs

Orthogonal optical injection in VCSELs was investigated for first time by Pan et al [2] who showed that an increase of injection strength can lead to bistable PS accompanied by injection locking. Recently, Altes et al [3] extended the work of Pan et al to experimentally map the dynamics of a VCSEL subject to orthogonal optical injection; in addition to PS, the results included regions of stable locking, bistability, subharmonc resonance, time-periodic and chaotic dynamics. Frequency-induced polarisation bistability for fixed injection strength has been investigated experimentally by Hong et al [26] and theoretically by Gatare et al [27]. Sciamanna and Panajatov [28] have used the SFM to analyse the case of orthogonal optical injection with similar values for fundamental parameters to those used in [5, 27] (γ_s=50 ns^-1), and Valle et al [29] have included the effects of the first-order mode as well as the fundamental in their SFM analysis of orthogonal injection (in this case with γ_s=91 ns^-1). Stability maps in terms of frequency detuning between master laser (ML) and slave laser (SL) for polarised optical injection in VCSELs, calculated using the SFM, have been reported [8,9,28]. In [8, 9] the spin relaxation rate is 40 ns^-1, and the birefringence rates are 9 ns^-1 and 5 ns^-1, whereas in [28] γs=50 ns^-1 and γ_p=30 ns^-1 with the exception of Fig. 17 where spin relaxation rates of 100, 200 and 300 ns^-1 are used to show the effect on the stability boundaries. In the next section we discuss further details of the EP and LP regions of the stability map and investigate the effect of varying these fundamental parameters.

7. Polarisation effects in the stability map of optically-injected VCSELs

7.1 SFM for polarised optical injection in VCSELs

By using y-polarised injection the equations for E_x, N and m are the same as (2), (4) and (5), whilst the equations for E_y, ϕ_y and ϕ_x become (see reference [8])

\frac{{dE}_{y}}{dt} = \frac{1}{2 τ_{p}} [(N - 1) E_{y} + m E_{x} \frac{\sin (ε - Δ ϕ)}{\cos ε}] + γ_{a} E_{y} + K_{inj} E_{inj} \cos Δ

\frac{d ϕ_{y}}{dt} = \frac{α (N - 1)}{2 τ_{p}} - \frac{m}{2 τ_{p}} \frac{E_{x}}{E_{y}} \frac{\cos (ε - Δ ϕ)}{\cos ε} - Δ ω_{y} - γ_{p} + 2 α γ_{a} + K_{inj} \frac{E_{inj}}{E_{y}} \sin Δ

\frac{d ϕ_{x}}{dt} = \frac{α (N - 1)}{2 τ_{p}} + \frac{m}{2 τ_{p}} \frac{E_{y}}{E_{x}} \frac{\cos (ε + Δ ϕ)}{\cos ε} - Δ ω_{y} - γ_{p}

where Δ=(αγ_a - γ_p)t - ϕ_y, K_injE_inj is the strength of optical injection and Δω_y is the detuning between the frequency of the ML and the y-LP mode of the SL. If the solitary laser is in LS_x (LS_y) the injection is considered as orthogonal (parallel). In our previous work [8] it was shown that the stability map for parallel injection shows no EP effects and the LP results are similar to those for conventional edge-emitting lasers. Wieczorek et al [16] give a comprehensive review of nonlinear dynamics in such lasers, using bifurcation theory to find the boundaries of chaos, limit cycle and injection locking. These features can also be found by calculating the largest Lyapunov exponent (LLE) [30], and this method also provides a means to measure the strength of the chaos.

In section 7.2–7.4 we will use orthogonal injection for the cases where the solitary VCSEL is in LS_x or ES. Previously we have discussed [8,9] the locking bandwidth, saddle node bifurcation and EP behaviour for limit cycle and chaos, named coupled limit cycle (CLC) and coupled chaos (CC). In the following sections we will extend that discussion, emphasising novel phenomena such as QS and details of elliptical injection locking stability (locking for E_x and E_y simultaneously) as well as the effect on the stability map of varying the fundamental parameters used in the SFM.

7.2 Fast spin relaxation and small birefringence and pumping

Choosing parameter values of 2η=1.3, γ_p=9rad/ns, γ_a=0 and γs=40 ns^-1 the polarisation of the solitary VCSEL is in the LS_x region of Fig. 2. The stability map for y-polarised injection with K_inj=120 ns^-1 is shown in Fig. 8. In addition to the three LP regions of chaos (black), limit cycle (white) and injection locking stability (light grey) the map of Fig. 8 includes the EP phenomena of coupled chaos (CC), coupled limit cycle (CLC), elliptical injection locking stability (EILS) and quasi stability (QS) denoted by light blue, dark blue, orange and dark grey colour coding respectively. Table 3 gives the colour coding for each of EP and LP phenomena and the corresponding oscillation frequencies of the dynamic variables for the cases of QS and limit cycles. QS behaviour is characterised by sinusoidal oscillations of very small amplitude around a fixed mean value for each of the dynamic variables of the system. In the case of the variables N and m for QS in the dark grey area of Fig. 8, the mean values are 1 and 0, respectively. A good approximation for the angular frequency Ω=Δϕ/t [8] of the variable m can be found by using these mean values in equations (17) and (18), to yield Ω=2γ_p+Δω_y. The remaining variables have frequency 2Ω. Inside the region QS at a detuning of Δω_y=-2γ_p=18 rad/ns, the oscillation frequency Ω tends to zero and the corresponding orange line denotes EILS where all variables have constant values. The main difference between QS and limit cycle is the limit cycle is characterised by in-phase periodic behaviour of the field components at the ROF (~2 GHz for the parameters used in Fig. 8) but the QS is characterised by sinusoidal behaviour with frequency related to the detuning and birefringence only, and the phase difference between field amplitudes is always a constant value as 2tan^-1(α) [8].

Fig. 8. Calculated stability map for parameters given in Table 1 and 2η=1.3, γ_s=40ns^-1, γ_p=9 rad/ns and K_inj=120 ns^-1. At Δω_y=-2γ_p=-18 rad/ns EILS is found.

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Table 3. Colour coding of EP and LP regions

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7.3 Slow spin relaxation with intermediate birefringence and pumping

As mentioned in section 4, with reduced spin relaxation and increased birefringence or pumping rates the ES region of a free-running VCSEL is enlarged. Hence, to investigate more details of EP phenomena, parameter values of 2η=1.5, γ_p=18 rad/ns, γ_s=20 ns^-1 (inside the LS_x area of Fig. 1), and K_inj=200 ns^-1 are used for the calculated stability map of Fig. 9. Here two more kinds of QS are found compared with Fig. 8. In the light green QS region all variables exhibit sinusoidal oscillations of very small amplitude around a fixed mean value with frequency Ω=2γ_p + Δω_y. The purple region of hybrid QS has a green QS behaviour for E_x, m and N, but E_y has a limit cycle behaviour with a frequency close to the ROF. In Fig. 9 there is another new area (dark green) of period-doubling behaviour that lies above the LLC.

Fig. 9. Calculated stability map for parameters given in Table 1 and 2η=1.5, γ_s=20 ns^-1, γ_p=18 rad/ns and K_inj=200 ns^-1.

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7.4 Slow spin relaxation with high birefringence and pumping

Figure 10 shows the stability map for y-polarised injection of strength K_inj=300 ns^-1 into a VCSEL whose solitary state lies in the region of ES with parameters 2η=2.1, γ_p=33.5 rad/ns and γ_s=20 ns^-1 (the ψ point of Fig. 3). The EP and LP regions are similar to those of Figs. 8 and 9, but the orange EILS is not based only on Δω_y=-2γ_p (Ω=0), as the elliptical stability occupies a relatively wide region. It can be concluded that using a solitary VCSEL in ES is a good route to EILS with y-polarised injection.

Fig. 10. Calculated stability map for parameters given in Table 1 and 2η=2.1, γ_p=33.5 rad/ns, γ_s=20 ns^-1 and K_inj=300 ns^-1.

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According to Figs. 8–10, at low values of injection level the entire map is dominated by QS behaviour. With increasing injection strength, the K_injE_injcosΔ and (K_injE_injsinΔ)/E_y terms of (16) and (17), respectively, cause different types of EP and LP behaviour in the map. Birefringence, as the coupling term between E_x and E_y, has the most important influence, compared to the other parameters, on QS behaviour. With increased pumping and birefringence the LP regions shift towards higher values of injection level (IL) and from positive to negative detuning. Polarisation switching can be achieved by injecting a y-polarised field into a solitary VCSEL in LS_x in Figs. 8 and 9 (and ES in Fig. 10); the stability map gives the operating point for the IL and detuning needed for switching.

7.5 Quasi-stability and quasi-periodicity

Figure 8 shows that the dark grey QS dominates most of the map for orthogonal injection in a relatively small detuning range (-20 rad/ns to 20 rad/ns). This dominance is maintained at higher values of detuning. For example, using the same parameter values as in Fig. 8 and an operating point at (IL,Δω_y)=(-40 dB, 70 rad/ns) the characteristic behaviour of the dynamic variables is shown in Fig. 11. Even if the detuning range is increased to extend from -80 rad/ns to 80 rad/ns, no effects of green QS or hybrid QS are found for the parameters of Fig. 8. A green QS region is achieved by using an operating point in Fig. 10 at (IL,Δω)=(-40 dB, 70 rad/ns). Figure 12 indicates that all variables have the same frequency approximated by Ω=2γ_p+Δω_y and the mean value of m is non-zero, unlike the dark grey QS where only m has this frequency with the frequency of other variables at twice the value, whilst the mean value for m is zero.

Fig. 11. Transient behaviour of the dynamic variables for parameters E_y/E_sol=0.01, γ_p=9 rad/ns, (IL,Δω)=(-40 dB, 70 rad/ns). The mean value for m is zero and Ω=87 rad/ns.

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Fig. 12. Transient behaviour of the dynamic variables for parameters E_y/E_sol=0.01, γ_p=33.5 rad/ns, (IL,Δω)=(-40 dB, 70 rad/ns). The mean value for m is non-zero and Ω=137 rad/ns.

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The elliptical quasi-periodicity (EQP), which is found in regions below the saddle node bifurcation, is a phenomenon with characteristic behaviour between chaos and limit cycle. Figure 13 is an example of EQP for an operating point of Fig. 10 (IL,Δω)=(-30 dB, -70 rad/ns).

Fig. 13. Transient behaviour of the dynamic variables for parameters E_y/E_sol=0.0316 and an operating point from Fig. 10 (IL,Δω)=(-30 dB, -70 rad/ns).

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8. Conclusions

In this paper the spin-flip model (SFM) equations have been used to study the polarisation behaviour of vertical-cavity surface-emitting lasers (VCSELs) for a typical range for pumping, birefringence and spin relaxation rates. By varying these parameters the characteristic behaviour of elliptical stability (ES) in a solitary VCSEL, such as the variation of field components and carrier densities with birefringence rate and pumping, has been calculated. In addition, for the first time to the best of our knowledge, the elliptical relaxation oscillation frequency has been analysed and compared with the linear relaxation oscillation frequency in SFM. From the sources of different behaviour in solitary VCSELs, we analysed the effects of varying fundamental parameters on the stability map of such VCSELs subject to polarised optical injection. Reducing the spin relaxation, or increasing the birefringence and pumping terms, can increase the ES for solitary and elliptical injection locking stability for slave VCSELs. The most significant aspect of the stability map calculated by the SFM for orthogonal injection compared to that for edge-emitting lasers is caused by the strong influence of birefringence; in general the chaotic area is reduced and replaced by elliptical quasi-periodicity, coupled limit cycle and quasi stability.

Appendix A

In the case of linear polarisation when E_y (or E_x) and m are zero there will be just two equations corresponding to only one field amplitude, E_o, and carrier density, n; in other words the equations are of the conventional forms without taking account of polarisation[16]. By using small perturbations E²=E_o ²+δE² and n=n+δn in the rate equations, a small-signal analysis for δE² and δN gives two complex conjugate roots (eigenvalues) of the form [16] γ_d ± j2πν _RO, where

γ_{d} = \frac{1}{2} [γ + G_{n} γ_{p} (J - J_{th})]

ν_{RO} = \frac{1}{2 π} \sqrt{G_{n} (J - J_{th}) - {γ_{d}}^{2}}

γ_d is the damping rate and ν_RO is the relaxation oscillation frequency (ROF), which is equal to the ROF of linear polarization, ν_LRO, in LS_x and LS_y cases. The second term in the square root is often negligible, so that equation (A2) simplifies to ν_LRO≈[G_n(J-J_th)]^1/2/2π. Using G_n=[τ_p(n_th-n_o)]^-1, (8) and J_th=n_th γ in ν_LRO and (A1) gives equations (14) and (15), respectively.

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Parameter	Symbol and value
Linewidth enhancement factor	α=3
Photon lifetime	τ_p=2ps
Electron decay rate	γ=1 ns^-1
Gain coefficient	G_n=1.1×10^-12 m³s^-1
Carrier density at transparency	n_o=1.1×10²⁴ m^-3
Spin relaxation rate (Figs. 1, 3–7, 9–10,12–13)	γ_s=20 ns^-1
Spin relaxation rate (Figs. 2, 8, 11)	γ_s=40 ns^-1

Colour coding	Type of polarisation	Ω: Frequency for E_y, E_x, N and m
	Linear chaos
	Linear limit cycle: period doubling	Ω ≈ ROF/2
	Linear limit cycle (LLC)	Ω ≈ ROF
	Linear stability (injection locked)	Ω=0
	Elliptical stability (injection locked)	Ω=0
	Quasi stability (QS): m_mean=0	Ω≈2γ_P+Δω_y for m, 2Ω for other variables
	Quasi stability (QS): m_mean >0	Ω≈2γ_P+Δω_y for all variables
	Hybrid QS: m_mean > 0	Ω≈2γ_P+Δω_y for all except E_y, Ω_Ey ≈ ROF
	Coupled limit cycle (CLC)	Ω ≈ ROF
	Elliptical quasi-periodicity
	Coupled chaos (CC)

Parameter	Symbol and value
Linewidth enhancement factor	α=3
Photon lifetime	τ_p=2ps
Electron decay rate	γ=1 ns^-1
Gain coefficient	G_n=1.1×10^-12 m³s^-1
Carrier density at transparency	n_o=1.1×10²⁴ m^-3
Spin relaxation rate (Figs. 1, 3–7, 9–10,12–13)	γ_s=20 ns^-1
Spin relaxation rate (Figs. 2, 8, 11)	γ_s=40 ns^-1

Colour coding	Type of polarisation	Ω: Frequency for E_y, E_x, N and m
	Linear chaos
	Linear limit cycle: period doubling	Ω ≈ ROF/2
	Linear limit cycle (LLC)	Ω ≈ ROF
	Linear stability (injection locked)	Ω=0
	Elliptical stability (injection locked)	Ω=0
	Quasi stability (QS): m_mean=0	Ω≈2γ_P+Δω_y for m, 2Ω for other variables
	Quasi stability (QS): m_mean >0	Ω≈2γ_P+Δω_y for all variables
	Hybrid QS: m_mean > 0	Ω≈2γ_P+Δω_y for all except E_y, Ω_Ey ≈ ROF
	Coupled limit cycle (CLC)	Ω ≈ ROF
	Elliptical quasi-periodicity
	Coupled chaos (CC)

Analysis of SFM dynamics in solitary and optically-injected VCSELs

Abstract

1. Introduction

2. Theory

3. Values for key parameters

4. Elliptically and linearly polarised stability

5. Analysis of relaxation oscillation frequency in ES

6. Polarised optical injection in VCSELs

7. Polarisation effects in the stability map of optically-injected VCSELs

7.1 SFM for polarised optical injection in VCSELs

7.2 Fast spin relaxation and small birefringence and pumping

7.3 Slow spin relaxation with intermediate birefringence and pumping

7.4 Slow spin relaxation with high birefringence and pumping

7.5 Quasi-stability and quasi-periodicity

8. Conclusions

Appendix A

References and links

Cited By

Figures (13)

Tables (3)

Equations (20)

Optics Express

Below Critical Point
γ_p < γ_px-min	γ_px-min < γ_p < γ_py-max	γ_py-max < γ_p
LS_y	LS_x or LB	LS_x
ν_LRO	ν_LRO	ν_LRO
Above Critical Point
γ_p ≤ γ_py-max	γ_py-max < γ_p < γp_EP-min	γ_pEP-min≤ γ_p < γ_px-min	γ_px-min ≤ γ_p
LS_y	Unstable	EP	LS_x
ν_LRO	-	ν_ERO(γ_pEP-min) < ν_ERO(γ_p) < ν_LRO(γ_px-min)	ν_LRO