Dynamics of laterally-coupled pairs of spin-VCSELs
DDynamics of laterally-coupled pairs of spin-VCSELs.
M.P. Vaughan , ∗ H. Susanto , I.D. Henning , and M.J. Adams School of Computer Science and Electronic Engineering, University of Essex,Wivenhoe Park, Colchester CO4 3SQ, United Kingdom and Department of Mathematical Sciences, University of Essex,Wivenhoe Park, Colchester CO4 3SQ, United Kingdom (Dated: January 15, 2020)A newly-developed normal mode model of laser dynamics in a generalised array of waveg-uides is applied to extend the spin-flip model (SFM) to pairs of evanescently-coupled spin-VCSELS. The effect of high birefringence is explored, revealing new dynamics and regionsof bistability. It is shown that optical switching of the polarisation states of the lasers maybe controlled through the optical pump and that, under certain conditions, the polarisationof one laser may be switched by controlling the intensity and polarisation in the other. ∗ Corresponding author: [email protected] a r X i v : . [ phy s i c s . op ti c s ] J a n I. INTRODUCTION
Recent years have seen a growth of research interest in the nonlinear dynamics of arrays ofvertical cavity surface-emitting lasers (VCSELs) and in potential applications of these effects.Notable advances include work on parity-time symmetry and non-Hermiticity associated with thecontrol of gain and loss in neighbouring VCSEL cavities [1–4]. Progress has also been rapid inthe understanding of ultrahigh-speed resonances that offer the prospect of very high frequencymodulation of coupled VCSELs and nanolasers [5–8]. Additional insight into optical couplingbetween adjacent elements of a two-dimensional VCSEL array has been achieved by careful analysisof the effects of varying the injected current independently on each array element [9]. The couplingwas shown to provide extra optical gain for array elements and thus lead to additional outputpower of the array due to in-phase operation [9, 10], reduced thresholds of individual elements[9, 11] and even cause unpumped elements to lase [9].In almost all the above examples of recent progress, modelling of the array behaviour based oncoupled mode theory (CMT) has been used to explain experimental results and develop improvedunderstanding of fundamental effects. Conventional CMT describes only the amplitude and phaseof the electric field of the photons and the total concentration of the electrons. Whilst this isadequate for modelling many phenomena occurring in laser arrays, it cannot easily be adapted toinclude the effects of optical polarisation or electron spin that are often relevant in vertical cavitylasers. For this purpose, the spin flip model (SFM) [12] is well-established as the method of choice,and has been successfully extended to model mutually coupled VCSELs by adding delayed opticalinjection terms [13]. This approach has been successfully applied recently to proposed applicationsof mutually coupled VCSELs in secure key distribution based on chaos synchronization [14] andreservoir computing based on polarization dynamics [15].Spin-VCSELs, where the polarisation and dynamics can be controlled by the injection of spin-polarised carriers, have recently attracted considerable attention since very high-speed ( >
200 GHz)modulation has been demonstrated [16] by applying mechanical stress to increase the birefringence.In the present contribution we explore some of the dynamics predicted for coupled pairs of spin-VCSELs based on a newly-developed theoretical treatment [17] that extends the SFM to apply toVCSEL arrays. This approach, which uses normal modes rather than CMT, accounts accuratelyfor instantaneous coupling via evanescent fields or leaky waves. It is therefore able to model thedetails of the optical guidance in the spin-VCSELs and effects of varying the spacing between them,thus going beyond the description offered by adding optical injection terms to the conventionalSFM. The next section gives a brief summary of this treatment leading to a set of rate equations.Subsequent sections deal with results, discussion and conclusions.
II. DOUBLE-GUIDED STRUCTURE
In Ref. [17], a general set of rate equations for any number of coupled lasers with an arbitrarywaveguide geometry and any number of optical modes, including the polarisation was derivedfrom Maxwell’s equations and the optical Bloch equations. In this model, the geometry of thewaveguides is introduced through the introduction of overlap factors , defined byΓ ( i ) kk (cid:48) ≡ (cid:90) ( i ) Φ k ( r )Φ k (cid:48) ( r ) d r (1)where k and k (cid:48) label the modes, Φ k ( r ) is the spatial profile of the k th mode and the integral isover the ( i )th guide. In fact, (1) represents a simplified model for which the gain is assumed to beuniform over a guide and zero elsewhere. The mathematical model of Ref. [17] allows for a moregeneral treatment, although this would greatly increase the complexity of the numerical solution.In an earlier work [18], we showed that the dynamics of coupled lasers in slab guides could be verysensitive to these overlap factors and stressed their importance. FIG. 1. Circular guides of radius a and edge-to-edge separation d . In this work, we set a = 4 µ m and allow d to be variable. In the present work, we consider the particular case of double-guided structures consisting oftwo identical circular guides of radius a = 4 µ m, as illustrated in Fig. 1. Note that in this paper,we take the edge-to-edge separation to be d (rather than 2 d as in Ref. [17]). We choose valuesof the cladding refractive index n and the refractive index in the guides n such that, for theoperating wavelength of λ = 1 . µ m, there are only two supported modes with even (for the lowerorder mode) and odd parity. We shall refer to these as the symmetric and anti-symmetric modesand denote them by k = s, a respectively. The values we choose are n = 3 . n = 3 . i ), Γ ( i ) ss is the overlap of the symmetricmodes, Γ ( i ) aa the overlap of the antisymmetric modes and Γ ( i ) sa is the cross product. Note that, dueto the symmetry of the guides, we always have Γ (1) sa = − Γ (2) sa , due to the parity of the modes.Moreover, as the separation d between them increases, we have Γ ( i ) sa → ( i ) ss → Γ ( i ) aa → Γ S / S is the optical confinement factor of an isolated guide. The factor of 1/2 arises since themodes are normalised over all space, which includes 2 guides.
1. Normalised rate equations
The general form of the normal mode model and its reduction to the double-guided structure indimensional and normalised form are derived in Ref [17]. Here we shall just quote the normalisedform used in our numerical calculations. The model has 11 independent variables: the spin-polarised carrier concentrations in each guide M ( i ) ± , where i ∈ { , } labels the guide and + / − labels the spin up / down components respectively; the optical amplitudes in each guide A i, ± ,where + / − labels the right-circularly / left-circularly polarised components respectively and threephase variables φ , φ −− and φ − . The φ ±± are the phase differences between A , ± and A , ± , which we shall refer to as the spatial phase . This is the phase of the coupled mode modelof Ref [19]. The variable φ − is the phase difference between A , + and A , − , which is the phase FIG. 2. A 3D schematic of two coupled circular waveguides encapsulating the essence of the applicationto a pair of VCSEL cavities. Shown are the cylindrical waveguide regions incorporating the active areas.Pumping is assumed to be confined to these regions. Note that we have omitted the Bragg stack mirrorsand substrate from this figure. referred to in the literature of the SFM. We shall call this the polarisation phase . A fourth phasevariable φ − is related to the other three via φ − = φ − φ −− + φ − .Note that the A i, ± are not the amplitudes of the modal solutions of the Helmholtz equationbut rather ‘composite modes’ defined in terms of a superposition of the actual modal solutions(symmetric and anti-symmetric) to better exploit the symmetry of the waveguide. Specifically,these become the amplitudes of the local solutions in isolated guides as the separation betweenthem is increased to infinity, retaining close similarity at nearer distances. Hence, they offer a moreintuitive, physical representation of the optical field in each guide. The actual normal modes maybe reconstructed from the composite modes and the phases using the procedure described in theappendix of Ref [17].For convenience of formulation in the double-guided structure model, we introduce new Γ termsdefined in terms of the optical overlap factors byΓ ( i ) ± = Γ ( i ) ss + Γ ( i ) aa ± ( i ) sa ( i ) = Γ ( i ) ss − Γ ( i ) aa . (3)Using these, we introduce further new variables defined via M ± = Γ (1)+ M (1) ± + Γ (2)+ M (2) ± Γ S , (4) M ± = Γ (1) − M (1) ± + Γ (2) − M (2) ± Γ S (5)and ∆ M ± = ∆Γ (1) M (1) ± + ∆Γ (2) M (2) ± Γ S , (6)in terms of which the optical rate equations are more concisely written.The normalised carrier rate equations are ∂M ( i ) ± ∂t = γ (cid:104) η ( i ) ± − (cid:16) I ( i ) ± (cid:17) M ( i ) ± (cid:105) − γ J (cid:16) M ( i ) ± − M ( i ) ∓ (cid:17) , (7)where η ( i ) ± are the polarised pumping rates in each guide, γ = 1 /τ N is the inverse of the carrierlifetime τ N , γ J is the spin relaxation rate and the polarised components of the optical intensity ineach guide are given by I ( i ) ± = Γ ( i )+ Γ S | A , ± | + 2 ∆Γ ( i ) Γ S | A , ± || A , ± | cos( φ ±± )+ Γ ( i ) − Γ S | A , ± | . (8)Note that in the normalised form of the SFM, the effective spin relaxation rate γ s = γ + 2 γ J isoften used.The normalised optical rate equations are ∂ | A , ± | ∂t = κ ( M ± − | A , ± | + [ κ ∆ M ± (cos( φ ±± ) − α sin( φ ±± )) − µ sin( φ ±± )] | A , ± |− [ γ a cos( φ − ) ± γ p sin( φ − )] | A , ∓ | , (9) ∂ | A , ± | ∂t = κ ( M ± − | A , ± | + [ κ ∆ M ± (cos( φ ±± ) + α sin( φ ±± )) + µ sin( φ ±± )] | A , ± |− [ γ a cos( φ − ) ± γ p sin( φ − )] | A , ∓ | , (10) ∂φ ±± ∂t = κα ( M ± − M ± ) + µ cos( φ ±± ) (cid:18) | A , ± || A , ± | − | A , ± || A , ± | (cid:19) + κ ∆ M ± (cid:20) α cos( φ ±± ) (cid:18) | A , ± || A , ± | − | A , ± || A , ± | (cid:19) − sin( φ ±± ) (cid:18) | A , ± || A , ± | + | A , ± || A , ± | (cid:19)(cid:21) + γ p (cid:20) cos( φ − ) | A , ∓ || A , ± | − cos( φ − ) | A , ∓ || A , ± | (cid:21) ∓ γ a (cid:20) sin( φ − ) | A , ∓ || A , ± | − sin( φ − ) | A , ∓ || A , ± | (cid:21) , (11)and ∂φ − ∂t = κα ( M − M − ) + µ (cid:18) cos( φ ) | A , + || A , + | − cos( φ −− ) | A , − || A , − | (cid:19) + κ ∆ M + ( α cos( φ ) + sin( φ )) | A , + || A , + | − κ ∆ M − ( α cos( φ −− ) + sin( φ −− )) | A , − || A , − | + γ a sin( φ − ) (cid:18) | A , + || A , − | + | A , − || A , + | (cid:19) + γ p cos( φ − ) (cid:18) | A , + || A , − | − | A , − || A , + | (cid:19) (12)The parameters of the optical model are the linewidth enhancement factor α , the cavity loss rate κ , the dichroism rate γ a , the birefringence rate γ p and the coupling coefficient µ . Note that µ isgiven in terms of the modal frequencies by [17, 20] µ = ν s − ν a , (13)for the symmetric ( s ) and anti-symmetric ( a ) modes found from solution of the Helmholtz equationfor the waveguiding structure.It will be convenient to define the pump ellipticity in the i th guide in terms of the right and leftcircular polarised pumping rates η ( i )+ and η ( i ) − by P ( i ) = η ( i )+ − η ( i ) − η ( i )+ + η ( i ) − . (14)Similarly, we may define the output optical ellipticity in the ( i )th guide via ε ( i ) = | A i, + | − | A i, − | | A i, + | + | A i, − | . (15)We describe this as the ‘modal’ ellipticity since it is in terms of the composite mode amplitudes.Although this is defined for each guide, there is a spatial dependence beyond this. The actualellipticity we would measure is given in terms of the spatially dependent components of the opticalintensity via ε ( x, y ) = I + ( x, y ) − I − ( x, y ) I + ( x, y ) + I − ( x, y ) , (16)where the I ± ( x, y ) are given in terms of the normal mode amplitudes A k, ± by I ± ( x, y ) = | A s, ± Φ s ( x, y ) + A a, ± Φ a ( x, y ) | . (17)Here, we have used the subscript k to distinguish these as the amplitudes of the modal solutionsof the waveguide (i.e. the solutions of the Helmholtz equation) rather than the ‘composite modes’used elsewhere in this paper (as discussed above), which are denoted by the subscript i . See Ref [17]for further details of this calculation. III. RESULTS AND DISCUSSIONA. Stability boundaries
The dynamics of pairs of laterally-coupled lasers with circular guides of radius a = 4 µ m havebeen investigated by plotting stability boundaries in the η ( i ) − d plane, where η ( i ) = η ( i )+ + η ( i ) − is the total normalised pumping rate in either guide and d is the edge-to-edge guide separation(Fig. 3). These plots are topologically equivalent to the scheme of Λ / Λ th v d/a diagrams used inRefs [18] and [17], where Λ and Λ th are the total pump power and threshold pump respectively.Here, because we may vary the pump ellipticity in each guide independently, Λ th is not well definedand so represents an inaccurate measure. FIG. 3. Stability boundaries in the η ( i ) = η ( i )+ + η ( i ) − verses edge-to-edge distance d plane for circular guidesequally pumped with a pump ellipticity of P ( i ) = 0 . γ p . A similar stability map, in terms of Λ / Λ th v d/a has been shown for the non-polarised case usingthe coupled mode model in Fig. 6 of Ref [19]. A remaining discrepancy between the results of thecoupled mode model and the present work is due to the sensitivity of the dynamics to the overlapfactors. It was shown in Ref [18] that, taking the asymptotic values of the overlap factors as theguide separation tended to infinity, the stability map for the non-polarised case reproduced thatof the coupled-mode treatment in Ref [19] exactly. This would then correspond to a birefringencerate of γ p = 0 ns − , which is almost indistinguishable from the case of γ p = 10 ns − plotted inFig. 3.For all the stability boundaries investigated here, we keep the total normalised pumping rate η the same in each guide and so may be conveniently plotted in the η ( i ) − d plane. In the regions ofinstability, we typically see oscillatory behaviour of the type reported in Section III B 3.In Ref. [17] stability boundaries were plotted for devices with a small birefringence rate γ p of0 FIG. 4. Stability boundaries for γ p = 30 ns − in the η ( i ) = η + + η − verses edge-to-edge distance d plane for circular guides equally pumped with a pump ellipticity of P ( i ) = 0 .
0. The solid lines are for the(polarisation) in-phase solutions and the dashed for the out-of-phase solutions. Stable in-phase solutions lieabove the solid line, whilst stable out-of-phase solutions lie to the right and beneath the dashed line. Theshaded area indicates the regions of bistability where both types of solution are stable. − , which gives very little coupling between the right and left circularly polarised componentsof the optical field. These gave rise to Hopf bifurcations qualitatively similar to the curve for γ p = 10 ns − shown in Fig. 3, up to around d = 25 µ m (in these calculations, all other parametershave been kept the same as in Ref. [17] for the purposes of comparison). In this earlier work,the stable, steady state solutions found above the curve were termed ‘out-of-phase’ solutions, inkeeping with the terminology of the coupled mode model [19]. In terms of the normal mode model,such out-of-phase solutions correspond to the anti-symmetric normal modes (at large separation,these tend to the solutions of isolated guides with a phase difference of π between them, meaningthe amplitudes are inverted). This phase relation is associated with the φ ±± variables, i.e. atlarge separation φ ±± = π . Earlier, we designated this the spatial phase to distinguish it from the1 TABLE I. Parameters used in numerical simulations.Parameter Value Unit Description α -2 Linewidth enhancement κ
70 ns − Cavity loss rate γ − Carrier loss rate γ a − Dichroism rate γ s
100 ns − Effective spin relaxation rate N . × cm − Transparency density a diff . × − cm − Differential gain n g γ s = γ + 2 γ J to aid direct comparison with the SFMmodel. These are the same parameters as used in Ref. [17] except that in this paper we vary thebirefringence rate γ p . polarisation phase associated with the φ ii + − variables.The graphs in Fig. 3 are calculated for pump ellipticities of P ( i ) = P (1) = P (2) = 0 (i.e. forlinearly polarised pumps). In Ref [17] it was shown that, in general, the stability boundariestended to move towards the origin as the pump ellipticity moves away from zero. In this work, weinvestigate the effect of increasing the birefringence rate γ p , which has the effect of coupling powerbetween the opposite circular components of the optical polarisation. Here, we see the emergenceof a new stability boundary moving roughly horizontally across the plane and increasing in η as γ p increases. These boundaries are plotted for the polarisation in-phase solutions, for which φ ii + − isclose to zero. These are characterised by the fact that the output optical ellipticity takes the samesign as the pump ellipticity. On the other hand, the ellipticity of the polarisation out-of-phasesolutions, for which φ ii + − is close to π , has the opposite sign to the pump ellipticity.Stability boundaries for both in-phase and out-of-phase solutions for for γ p = 30 ns − and valuesof P ( i ) = P (1) = P (2) from 0 to 0.8 are shown in Figs 4 to 8 (from here on, we shall be referringto the polarisation phase whenever we speak of in-phase or out-of-phase solutions without specificqualification). These show the out-of-phase stability boundaries as dashed lines with the stablesolutions to the right of the curved borders and beneath the horizontal borders. Investigating the2 FIG. 5. Stability boundaries for γ p = 30 ns − in the η ( i ) = η + + η − verses edge-to-edge distance d planefor circular guides equally pumped with a pump ellipticity of P ( i ) = 0 .
2. The grey lines show continuationsof the Hopf bifurcation into the unstable region. Note that, unlike the other stability maps shown, in thiscase there is no region of bistability. sharp kinks in the borders, we find that this is due to the continuation of Hopf bifurcations intothe unstable regions. An example is shown in Fig. 5 in the case of P ( i ) = 0 .
2A clear feature of these stability boundaries is that, in most cases, the out-of-phase boundarycrosses that of the in-phase boundary creating regions of bistability where both types of solutionare stable. These are shown as the shaded areas and suggest the possibility of optical switchingbetween these stable states. This is investigated in the next sub-section, where we find that opticalswitching via pump power and / or ellipticity is indeed achievable.In the case of P ( i ) = 0 .
2, we note that, unlike the other cases shown, there is no region ofbistability in domain plotted. At this point, however, we can offer no definitive explanation forthis behaviour.3
FIG. 6. Stability boundaries for γ p = 30 ns − in the η ( i ) = η + + η − verses edge-to-edge distance d planefor circular guides equally pumped with a pump ellipticity of P ( i ) = 0 . B. Bistability
1. Switching both lasers together on pump power
We have examined the dynamics within the bistable regions via time series solutions of therate equations using the Runge Kutta method (technical details are given in Ref. [17]). Each timeseries is run for a simulation time of 400 ns for a given pump power and ellipticity in each guide.The output solutions at the end of each solution are then used as the intial conditions for the nextsimulation with different pumping parameters. In this way, we can see how the system behaves aswe vary these parameters smoothly or in sharp jumps.For an initial set of simulations, we keep the birefringence at γ p = 30 ns − , take the edge-to-edgeseparation to be d = 20 µ m and the pump ellipticity in either guide to be P ( i ) = P (1) = P (2) = 0 . FIG. 7. Stability boundaries for γ p = 30 ns − in the η ( i ) = η + + η − verses edge-to-edge distance d planefor circular guides equally pumped with a pump ellipticity of P ( i ) = 0 . power η ( i ) = η (1) = η (2) = η ( i )+ + η ( i ) − = 12. From Fig. 7 we can see that this is in a region ofinstability for the in-phase solution but just on the edge of the stable region for the out-of-phasesolution. We then start increasing the pump power in both guides and track the modal outputoptical ellipticity ε ( i ) , given by (15). This is shown in Fig. 9, where at η ( i ) = 12 we have ε ( i ) = − . η ( i ) = 22, ε ( i ) = − .
44 following the direction of the red solid arrow. After thispoint, we enter into a region of unstable dynamics where the system fails to settle down to thein-phase steady state solution until the power reaches η ( i ) = 56. This is indicated by the dashedred arrow. At this point, we track back, ramping down the power. This time, the system remainsin the in-phase steady state solution all the way through the bistable region until it cross the Hopfbifurcation delimiting the in-phase dynamics and the system drops to the out-of-phase solution.It is natural to ask whether we may obtain switching behaviour by applying step changes tothe pump. To investigate this, we start the system off in an out-of-phase steady state solution5 FIG. 8. Stability boundaries for γ p = 30 ns − in the η ( i ) = η + + η − verses edge-to-edge distance d planefor circular guides equally pumped with a pump ellipticity of P ( i ) = 0 . with η ( i ) = 22 in both lasers. This gives an output ellipticity of ε ( i ) = − .
44. We then step upthe power to η ( i ) = 56 for a period of 20 ns. This settles down to a steady-state in-phase solutionwith ε ( i ) = 0 . η ( i ) = 22. However, the system now settles down in an in-phase steady-state with ε ( i ) = 0 . η ( i ) = 22 and the system switches to an out-of-phase solutionwith ε ( i ) = − .
2. Finally, stepping the power back up to η ( i ) = 22, we arrive back at the out-of-phase solution with ε ( i ) = − .
44. Hence, we can use the pump power for the purposes of opticalswitching, with an overal switching time of around 20 ns in this case (giving a possible switchingrate of around 8 MHz).The switching dynamics are explored in more detail in Figs. 11 to 14 on the sub-nanosecondtime-scale. Fig. 11 shows the dynamics as the system is switched from the out-of-phase solutionat η ( i ) = 22 to the in-phase at η ( i ) = 56. We see on this scale that the behaviour is oscillatory,6 FIG. 9. Hysteresis curve of the ellipticity for equally pumped guides with γ p = 30 ns − , an edge-to-edgeseparation of d = 20 µ m and pump ellipticity P ( i ) = 0 .
6. The points trace out the dynamics as the totalpump power η is changed gradually in the direction of the arrows. varying between around ε ( i ) = − . ε ( i ) = 0 . · ns − ( ∼
10 GHz). Figs. 12 to 13 show the steps from η ( i ) = 56 to η ( i ) = 22 and η ( i ) = 22 to η ( i ) = 12 respectively on the same scale, with similar angular frequencies of 72 rad · ns − ( ∼
11 GHz)and 66 rad · ns − ( ∼
11 GHz). In the final step from η ( i ) = 12 to η ( i ) = 22 shown in Fig. 14, thesystem settles down much faster. The angular frequency of the oscillations in this case is around58 rad · ns − ( ∼ ω R ofdamped oscillations given in Ref. [19] derived from a stability analysis of the coupled mode model ω R = 2 γκ ( η − − γ D , (18)7 FIG. 10. Time series showing the response of equally pumped guides with an edge-to-edge separation of d = 20 µ m and pump ellipticity P ( i ) = 0 .
6, to stepped total pump powers of η ( i ) = 22 , , ,
12 and 22again. This demonstrates a mechanism of switching between two stable solutions for η ( i ) = 22 with opticalellipticities of the opposite sign (indicated by the dashed lines). where γ D is the damping rate given by γ D = − γη . (19)However, for values of η = 56 ,
22 and 12, using (18) we obtain values of ω R = 83 ,
53 and 39 ns − respectively, showing a strong dependence on the pump power η .Instead, we note that in the analysis of spin-polarised VCSELS [16, 21, 22], it has been foundthat the frequency of birefringence-induced oscillations was mainly determined by the birefringencerate γ p , given approximately by γ p /π for large γ p (in GHz if γ p is given in ns − ). In our case, wehave γ p = 30 ns − , giving γ p /π = 9.5 GHz, which is very close to the observed frequency in thenumerical simulations.8 FIG. 11. Detail of Fig. 10 between t = 19 . t = 21 . η ( i ) = 22. As the pump power in each guide is stepped up η ( i ) = 56, the ellipticityinitially undergoes oscillations with an angular frequency of approximately 64 rad · ns − before settling downto a stable steady state in-phase solution.
2. Switching one laser via the other
Having verified that is possible to switch the ellipticity of the lasers in the bistable region byvarying the pump powers in each simultaneously, we next investigate the possibility of switchingone laser purely by varying the pump on the other, hence via the coupling between them. Thefollowing is a proof of concept and is not supposed to represent the optimal conditions for suchfunctionality.The edge-to-edge separation is taken to be a little shorter at d = 18 µ m and for the initialinvestigation, the total pump power in either guide is held fixed at η ( i ) = 20. The birefringenceis γ p = 30 ns − as before. Initially, the pump ellipticity is set at P ( i ) = 0 . P (1) is kept fixed9 FIG. 12. Detail of Fig. 10 between t = 39 . t = 41 . η ( i ) = 56, the pump power in each guide is stepped down to η ( i ) = 22. The ellipticity initially undergoesoscillations with an angular frequency of approximately 72 rad · ns − and a much smaller amplitude than inFig. 11 before decaying to the constant in-phase solution for η ( i ) = 22. throughout and P (2) is then varied, initially being increased to P (2) = 1 and then reduced again to P (2) = 0 . P (2) = 0 .
5, both lasers drop to an out-of-phase steady-state solution, which thenvaries smoothly as P (2) is reduced to -0.7. During this variation, the laser in guide (1) remainsin an out-of-phase solution, whilst the ellipticity in guide (2) varies linearly from an out-of-phasesolution to an in-phase solution. Beyond P (2) = − .
7, neither laser settles to a steady-state.As P (2) = − . P (2) = 0 .
7, the ellipticity tracks back over its previous valuesand then continues to vary smoothly past the point where the in-phase solution dropped to theout-of-phase solution. These behaviours are shown in Fig. 15 where the square points show theellipticity in guide (1), the diamond points show the ellipticity in guide (2) and the red arrowsindicate the directions in which P (2) is varied.0 FIG. 13. Detail of Fig. 10 between t = 59 . t = 61 . η ( i ) = 22,the pump power is stepped down to η ( i ) = 12, leading to oscillations in the ellipticity with an angularfrequency of approximately 66 rad · ns − . In this case the amplitude of the oscillation increases until it isvarying between -1 and 1 before suddenly collapsing to a steady state out-of-phase solution. In fact, it is found that once the system is on the lower line of Fig. 15 with guide (1) in anout-of-phase steady-state solution, it cannot be switched back to an in-phase state by varying P (2) .This can only be achieved by varying the pump power. However, it can be achieved by only varyingthe pump power in laser (2), so the goal of switching one laser purely by coupling with the otheris achievable.Specifically, we can use the following sequence: Starting with η ( i ) = 20 and P ( i ) = 0 . (cid:15) (1) = 0 .
19. Stepping P (2) to 0.4, (cid:15) (1) drops to -0.40. Putting P (2) backto 0.6, (cid:15) (1) changes very little, with (cid:15) (1) = − .
39. If we now step η (2) up to 60, the ellipticity inguide (1) then changes to (cid:15) (1) = 0 .
19. Dropping the power in guide (2) back down to 20, we endup again in the original in-phase solution with (cid:15) (1) = 0 .
19. For this particular set of parameters,the switching time is quite slow, taking around 100 ns to settle down to the steady-state solutions.1
FIG. 14. Detail of Fig. 10 between t = 79 . t = 81 . η ( i ) = 12, the pump power is stepped up to η ( i ) = 22. The ellipticity oscillates with anangular frequency of approximately 58 rad · ns − and then very rapidly decays to the out-of-phase steadystate solution.
3. Oscillations in the ellipticity
This switching behaviour on the basis of variation of P (2) does not occur under all conditionswithin a bistable region. At d = 20 µ m, the variation in the output ellipticities is similar to thatshown in Fig. 15 except that there is drop from the steady-state in-phase solutions to the out-of-phase steady-state solutions as P (2) is reduced. Instead, the system becomes unstable with theellipticity oscillating as shown in Fig. 16 for P (2) = 0 . T = 0 . P (2) is varied from 0.3 to -1, although the maximaand minima of the oscillations do. This variation is shown in Fig. 17. We note a qualitative breakin behaviour between P (2) = − . P (2) = − . FIG. 15. Optical ellipticity switching by varying the pump ellipticity in guide 2. Here γ p = 30 ns − , d = 18 µ m, P (1) = 0 . η ( i ) = 20. The grey squares show the optical ellipticity in guide 1 and the whitediamonds show the ellipticity in guide 2, for which the pump ellipticity was directly varied. The red arrowsshow the sequence in which the pump ellipticity in guide 2, P (2) , was varied. IV. CONCLUSIONS
A recently-developed theory of evanescently-coupled pairs of spin-VCSELs has been applied tostudy the dynamics of structures with two identical circular cylindrical waveguides and realisticmaterial parameters. Stability boundaries in the plane of total normalised pump power versusedge-to-edge spacing of the lasers have been presented for the cases of (1) zero pump polarizationellipticity with varying birefringence rate, and (2) fixed birefringence and varying pump ellipticity,with equal pump power in each laser for all cases. Boundaries for in-phase and out-of-phasesolutions are found in terms of the spatial phase of the normal modes of the system. It is shownthat intersection of these boundaries can give rise to sharp kinks in the overall stability boundariesfor some pump ellipticities, whilst for others crossing of the in-phase and out-of-phase solutions3
FIG. 16. Oscillations in the optical ellipticity for guide 1 (black) and guide 2 (grey) with a separation of d = 20 µ m. Here γ p = 30 ns − , P (1) = 0 . P (2) = 0 . η ( i ) = 22. can yield regions of bistability. The dynamics of the coupled spin-VCSELs in the bistable regionshave been examined by time series solutions of the rate equations. It is shown that it is possibleto switch the output ellipticity of the lasers by varying the pump powers in each simultaneously.It is also possible to switch the output elllipticity of one laser by varying the pump ellipticityor pump power of the other, under certain operating conditions. For other conditions, however,values of the pump ellipticity of one laser can be found that produce oscillatory behaviour of theoutput ellipticities of both lasers. Thus, it has been demonstrated that evanescently-coupled pairsof spin-lasers can yield a rich variety of different dynamics. Further work is needed to explore theeffects of varying material, device and operating parameters and hence to investigated potentialapplications of these dynamics.4 FIG. 17. Minima and maxima of the oscillations in the optical ellipticity for guide 1 (squares) and guide 2(diamonds) with a separation of d = 20 µ m as P (2) is varied. Here γ p = 30 ns − , P (1) = 0 .
6, and η ( i ) = 22. ACKNOWLEDGEMENT
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