Study of optical bistability based on hybrid-cavity semiconductor lasers

All-optical flip-flop has been demonstrated experimentally based on our optical bistable hybrid square-rectangular lasers. In this paper, dual-mode rate equations are utilized for studying the optical bistability in the two-section hybrid-coupled semiconductor laser. A phenomenological gain spectrum model is constructed for considering the mode competition in gain section and saturable absorption effect in the absorptive section in a wide wavelength range. The mechanisms of the optical bistability are verified in the aspect of the distribution of carriers and photons in the two-section hybrid-coupled cavity. In addition, we find that with the adjustment of the device parameters, both of the width and biasing current for achieving the bistability can be tuned for a wide range. Furthermore, a dynamic response of all-optical flip-flop is simulated, using a pair of set/reset optical triggering pulses, in order to figure out the laws for faster transition time with lower power consumption.All-optical flip-flop has been demonstrated experimentally based on our optical bistable hybrid square-rectangular lasers. In this paper, dual-mode rate equations are utilized for studying the optical bistability in the two-section hybrid-coupled semiconductor laser. A phenomenological gain spectrum model is constructed for considering the mode competition in gain section and saturable absorption effect in the absorptive section in a wide wavelength range. The mechanisms of the optical bistability are verified in the aspect of the distribution of carriers and photons in the two-section hybrid-coupled cavity. In addition, we find that with the adjustment of the device parameters, both of the width and biasing current for achieving the bistability can be tuned for a wide range. Furthermore, a dynamic response of all-optical flip-flop is simulated, using a pair of set/reset optical triggering pulses, in order to figure out the laws for faster transition time with lower power consumption.


I. INTRODUCTION
With the development of transmission and switching technologies, optical switching fabrics are greatly demanded for potentially reducing the complexity and power dissipation. 1Optical bistability based on saturable absorption, [2][3][4] two-mode competition, [5][6][7][8] and dispersion 9 have been extensively investigated for decades as they have potential applications in optical computing and fiber-opticbased telecommunication. 10Among them, optical bistability is first analyzed for an inhomogeneous current-injected laser based on a saturable absorber, 11 but the signal-processing speed is limited by the carrier lifetime on the reset edge. 3In contrast, bistability based on two-mode competition was more attractive in terms of optical switching time. 12,13[17][18][19] Recently, a hybrid square-rectangular laser (HSRL) has been designed and fabricated to realize stable single-mode operation. 20,21n addition, a 3-dB small-signal modulation bandwidth of 15.5 GHz and relative intensity noise almost reaching the standard quantum shot-noise limit are realized because of the strong mode selection of the square microcavity. 22Furthermore, controllable bistability and all-optical flip-flop are experimentally demonstrated due to mode competition and saturable absorption in the square microcavity section. 23In this study, rate equations for a two-section hybridcavity laser with dual-mode competition are constructed considering the nonlinear gain effect and saturable absorption, and static characteristics and dynamic transition responses are numerically simulated for faster transition speed and lower power consumption.The paper is organized as follows.In Section II, experimental results of a bistable laser are briefly summarized.In Section III rate equations are set up with a simplified gain spectrum model.The temperature dependence of the bistable loops is simulated in Section IV, which is in good agreement with the experimental results.In addition, device parameters are optimized for stable all-optical flip-flop with low power consumption.Furthermore, dynamic transition responses of all-optical flip-flop are investigated under moderate and strong mode-coupling situations in Section V and Section VI, respectively.Finally, the conclusion is drawn in Section VII.

II. EXPERIMENTAL RESULTS
HSRL, composed of a Fabry-Perot (FP) cavity and a square microcavity, is fabricated using an AlGaInAs/InP laser wafer with six compressively strained quantum wells. 20The conventional planar technology is utilized to fabricate lasers without additional epitaxial regrowth.After benzo cyclobutene planarization, an electrical isolation trench region is formed by inductively coupled plasma (ICP) etching the p-InGaAs contact layer off between the FP cavity and the square microcavity to guarantee mutual electrical isolation.Subsequently, two patterned p-electrodes and the n-type backside electrode are evaporated.The lasers are cleaved over the FP section to a length of about 300 µm to form a reflective facet, while the other side is a square microcavity, as shown in Fig. 1(a).With strong mode coupling between the square microcavity and the FP cavity, an HSRL can realize stable single-mode operation with a side-mode suppression ratio over 45 dB.Mounted on an AlN submount with the temperature controlled by a thermoelectric cooler (TEC), the fabricated HSRLs were tested with the FP cavity under a continuous-wave (CW) injection current as a gain section, while the square microcavity without an electrical pump worked as an absorber.The output power collected by a single-mode fiber versus the injection current of the FP section (IFP) at TEC temperatures of 289, 293, 296, and 300 K is measured and plotted in Fig. 1(b) for an HSRL with square side length a = 15 µm, FP cavity length l = 300 µm, and width w = 1.5 µm.A bistable loop around the threshold caused by the saturable-absorptive effect in the square microcavity is observed, just as in common-cavity two-section bistable lasers. 3In addition, a bistable loop due to the mode competition is realized when IFP ranges from 50 mA to 54 mA at 289 K.The lasing spectra measured by increasing and decreasing IFP are illustrated in Figs.1(c)-1(j) at a TEC temperature of 289 K and IFP = 49, 51, 53, and 55 mA.Lasing mode hopping between M1 at λ 1 = 1529 nm and M2 at λ 2 = 1560 nm is observed.The obtained mode interval of 31 nm is equal to two times of the longitudinal mode interval of the square microcavity, due to high coupling efficiency for FP mode with the symmetric mode in the square microcavity. 20And the fundamental transverse modes with symmetry distribution has an interval of two times of the longitudinal mode interval. 24As the temperature is just higher than 297 K, both bistable loops disappear suddenly owing to the red shift of the gain spectrum with the increase in the temperature.

III. RATE EQUATION MODEL A. Two-section dual-mode equations
To study the transition mechanism of the steady state and dynamic response of an HSRL, comprehensive rate equations including two electrically isolated sections and dual resonance modes within the gain spectrum are constructed with a nonlinear gain and a saturable absorptive process.Carrier densities are assumed to be uniform in each cavity, and carrier diffusion between cavities is neglected.Assuming the electrical field to be E 0 exp[j(ωt+ψ)], the rate equations for the carrier densities in the FP cavity and square microcavity and the average photon density for modes M1 and M2 at λp with p = 1 and 2 in the cavity are described as: 25,26 where N and s are the average carrier density and the photon density, respectively.Subscripts FP and SQ represent the parameters in the FP cavity and in the square microcavity, respectively; IFP is the injection current in the FP section; ηe is the current injection efficiency; qe is the electron charge; RFP and RSQ are the proportions of the active region volumes of the FP cavity and square microcavity over the whole device, respectively; A, B, and C are the defect, bimolecular, and Auger recombination coefficients, respectively; vg = c 0 /ng is the group speed of the lasing mode; and ng is the group refractive index.Γz is the optical confinement factor; the proportion of the photon number confined in the FP cavity to the whole photon number in the laser is assumed to be a constant γ. αint is the material internal absorption loss.β and α are the spontaneous emission factor and the linewidth enhancement factor, respectively.The parameter values are listed in Table I.The passive mode lifetime determined by the passive mode quality factor Qp and the resonant angular frequency ωp is

B. Rate equations with optical injection
All-optical flip-flop operation is simulated by injecting optical trigger pulses around the mode frequencies as set/reset signals with the FP section biased under a CW injection current within the hysteresis loop.Accounting optical injection item as κcEinj(t)exp (j∆ωt), the time-varying photon density and phase items are described as 26 where the frequency detuning and phase difference of the injected optical pulse at the frequency ωi relative to the free-running lasing modes are: respectively, while the angular frequency ωp and the phase p are assumed to be constant under a certain injection current and stable temperature.κc = √ ηo τpc is the injected light coupling coefficient, where ηo represents the optical injection coupling efficiency from the fiber to the active region of the device, which is considered to be 0.23 according to the experimental result. 23sinj,p(t) is the injected optical trigger signal sequence with the amplitude represented by sinj,p, whose shape is assumed to be rectangular.The injected pulse energy at λp is described as where ∆t is the injected optical trigger pulse width, h is Planck constant, and Va =VFP+VSQ is the active region volume of the whole device.
The gain coefficients gFP,p(NFP, λp, T, sp), and gSQ,p(NSQ, λp, T, sp) are related to the carrier density N in the FP cavity or in the square microcavity, average photon density sp, mode wavelength λp, and temperature T. A simplified gain model considering the saturable absorptive and the nonlinear gain effect is described in the following section.

C. Gain spectrum
The phenomenological gain spectrum is constructed based on the relation between the material gain spectrum and the spontaneous emission spectrum as described in Refs.27,28 where k is the Boltzmann constant, T is the absolute temperature, ∆F is the quasi-Fermi level separation energy, and n eff is the refractive index.The spontaneous emission spectrum Rsp is assumed to be a Gaussian function as with the half width and the peak wavelength of δλ and λ 0 .respectively.The broadening of the Rsp and radiative recombination coefficient B(N) with the carrier density N is taken into account as 28 with b = 2.5 × 10 −19 cm −3 , δλ 0 = 40 nm, B 0 = 2 × 10 −10 cm 3 s −1 , and B1 = 2.5 × 10 19 cm -3 .The peak wavelength λ 0 of Rsp is presumed to shift with the temperature at a rate of 0.54 nm/K. 29he peak wavelength λc of the gain spectrum can be obtained by taking the differential of (10) to be zero The gain coefficient at λc in the FP section is assumed to be a logarithmic function for the semiconductor material of multiquantum wells as in Ref. 30 where Ntr and Ns are the transparency carrier density and the logarithmic gain parameter, respectively.g 0 is the intrinsic material gain parameter presumed to be decreased at a rate of −10 cm −1 /K.By combining Eqs. ( 10)-( 15), the peak wavelength λc and the quasi-Fermi level ∆F can be obtained from the gain spectrum as the carrier density N is greater than 1.4 × 10 18 cm −3 .
To model the gain (absorption) spectrum in the square cavity under low carrier density, we first estimate the quasi-Fermi level approximately using the Boltzmann distribution under as low carrier density as N < 1 × 10 18 cm −3 by where the intrinsic carrier density Ni is given by mn and mp are the electron and hole effective masses and E g0 is the bandgap of the material.Based on the numerical results at N < 1 × 10 18 cm −3 and N > 1.4 × 10 18 cm −3 represented by the symbols in Fig. 2(a), we estimate the whole range of the ∆F as a function of N by fitting the numerical results with a third-order polynomial function.The obtained ∆F versus the carrier density N is plotted as a solid line in Fig. 2(a) at 280, 300, and 320 K. Substituting the obtained ∆F into (10), we can obtain the gain spectrum as a function of λ, N, and T. The environmental temperature is controlled by the TEC, and the heat effect is considered as a function of the injection current.
Considering the effects of carrier spatial and spectral hole burning, the gain spectrum with a carrier density larger than Ntr is described as gFP,p(NFP, λp, T, sFP,p) = g(NFP, λp, T) where εpp and εpq are the self-and cross-gain suppression factors, respectively, which are ε 11 and ε 12 for mode M1 at λ 1 and ε 22 and ε 21 for mode M2 at λ 2 .
Using the parameters listed in Table I, the material gain coefficient g is calculated as a function of the wavelength, carrier density, and temperature.As shown in Figs.2(b), the whole gain spectrum up shifts with the increase in the carrier density.With the increase in the temperature, the peak wavelength redshifts, and the peak gain decreases, as shown in Fig. 2(c), at a carrier density of 1.6 × 10 18 cm −3 .At fixed N = 1.0 × 10 18 cm −3 , the absorption coefficients decrease almost linearly with the temperature in the short-wavelength range, while reaching zero in the long-wavelength part, as shown in Fig. 2(d).

IV. STEADY-STATE SIMULATION RESULTS
In this section, the steady-state characteristics of the HSRL are investigated by solving Eqs. ( 1)-(3) using the gain spectrum model described in Section III.The temperature dependences of the bistable loops are calculated and compared with the experimental results.The output power coupled into the fiber of mode p can be expressed as where V eff = Va/Γz is the mode volume, σ is the output power coupling efficiency into a single-mode fiber, which is considered to be 0.5 and 0.45 for mode M1 and M2, respectively.The heating effect caused by the injection current is also considered.The practical laser temperature is fitted as a function of IFP (mA) as ∆T = (0.24IFP +0.0068IFP 2 ) K by assuming the mode wavelength versus the temperature at a rate of 0.10 nm/K. 33The calculated output powers as functions of IFP at TEC temperatures of 280, 285, 290, 295, and 305 K are illustrated in Fig. 3. Bistable loops are observed similar to the experimental results in Fig. 1(b), but the increase in the threshold current is relatively slow due to the neglection of temperature dependence of the Auger recombination. 34The heating effects in the square microcavity caused by the increase in IFP are neglected in the simulation.
The detailed output powers versus the IFP ranging from 47 mA to 59 mA at 295 K are plotted in Fig. 4 verify the bistable operation mechanism.The lasing mode is first located at M1 with a higher carrier density near absorption saturation at λ 1 .By further increasing IFP, the lasing mode is switched to mode M2 at λ 2 because of the redshift of the FP gain spectrum caused by the heating effect.The carrier density in the square section decreases because of the lower absorption at long-wavelength mode M2, as shown in Fig. 3(d).Inversely, with the decrease in the FP current, the mode gain around M1 increases and abrupt mode switching to M1 occurs as the absorption loss of M1 in the square section is compensated.The square section without electrical biasing provides a threshold gain difference required for optical bistability.In the case of no biasing, the carrier density in the square microcavity is smaller than the transparency carrier density at the pumping wavelength.A series of simple schematic diagrams with the gain and absorption spectra has been illustrated in Ref. 23.With further increase in the TEC temperature to greater than 298 K, both bistability regimes disappear, as shown in Fig. 3 at 305 K.In this case, the lasing mode starts at λ 2 owing to the redshift of the gain spectrum caused by the increase in the TEC temperature, which is in accordance with the experimental results in Fig. 1(b).
The bistability behaviors are also investigated under different values of RFP and γ for optimizing the hysteresis width.The output powers versus IFP are calculated and plotted in Fig. 5(a) as RFP = 0.2, 0.4, 0.6, 0.8 and γ = 0.4.With the increase in RFP, the threshold current decreases and the output power is enhanced owing to the increase of the FP section.Bistable loops at RFP = 0.7 are illustrated in Fig. 5(b) under γ =0.35, 0.4 and 0.6.Wide bistable loops around the threshold and the mode competition region are reached as γ reaches 0.35.With further increase in γ, the absorptive effect in the square section becomes weaker, and the bistable loops become narrower and tend to vanish with the increase in γ.

V. DYNAMIC RESPONSE SIMULATION RESULTS
By biasing the HSRL laser within the mode-competition bistable region at IFP = 62 mA and ISQ = 0 mA with an environmental temperature of 290 K, the dynamic transition responses are investigated by solving Eqs. ( 1)-( 2) and ( 5)-( 6) using the secondorder Runge-Kutta integration method with a time step and a time span of 100 fs and 10 ns, respectively.High-speed all-optical flip-flop operations are observed, as shown in Fig. 6, triggered by set/reset optical pulses with a pulse width of 100 ps, frequency detuning ∆f p in (7) of −10 GHz, and pulse energies of 47 fJ and 13 fJ for M1 and M2, respectively.As shown in Fig. 6, the photon density of the low-state mode increases immediately once an optical trigger pulse is injected, following with a dip right after the disappearance of the trigger signal due to the depletion of the carriers by the injected optical signal.Subsequently, the device undergoes oscillations with a frequency of several gigahertz before being stabilized at the steady-state value, which corresponds to the relaxation oscillation induced by a large carrier density deviation from the steady state.

VI. DYNAMIC RESPONSE UNDER STRONG NONLINEAR GAIN
Finally, strong nonlinear gain coefficients are considered for improving the switching properties.In the case of strong mode coupling, abrupt mode switching can be realized. 25Setting the selfgain suppression coefficients ε 11 =ε 22 = 8×10 -17 cm 3 and the crossgain suppression coefficients ε 12 =ε 21 = 20×10 -17 cm 3 , we simulate the transition responses for M1 and M2 triggered by an optical trigger pulse with a pulse width of 100 ps and pulse energies of 36 fJ and 6 fJ, as shown in Fig. 7. Dips at the rise/fall edges caused by optical triggering pulses still exist; however, the acute oscillation at the rise and fall edges are greatly suppressed, which is beneficial for the improvement of the signal processing speed and fidelity.The selfand cross-gain suppression coefficients are related to the spectral and spatial hole burning effect, which may dependent on the device structural parameter, material, 35 working conditions 36 and also the mode coupling.Their ratio can be modified by adjusting the mode intensity distribution overlapping by designing the structure of the square microcavity. 37However, the detailed controlling of the two coefficients still needs further investigation.
The dynamic transition responses of the HSRL with γ = 0.4, 0.5, and 0.6 are investigated and plotted in Fig. 8. Triggered by pulses with the same pulse width of 100 ps and energies of 36 fJ and 6 fJ around M1 and M2, respectively, both the rise time and turnon oscillation of the all-optical flip-flop device are improved as γ is increased from 0.4 to 0.6, especially for the transition edges for the mode switching from M1 to M2.The practical device parameter γ can be controlled by the mode Q factor of the square microcavity, which can be modified by varying the side length or introducing circular side for the square microcavity with optimal deformation amplitude. 37n addition, the pulse energy required to realize mode switching under a wide range of frequency detuning is investigated.The minimum input trigger energy required for mode transition is 14.2 fJ and 3.2 fJ at a frequency detuning of −10 GHz and −20 GHz for M1 and M2, respectively.Away from the optimal frequency detuning, the trigger energy required to realize mode switching increases

VII. CONCLUSIONS
In conclusion, we have established comprehensive two-section dual-wavelength rate equations with a gain spectrum considering the nonlinear gain and absorptive effect in a wide wavelength range.Based on the steady-solution of the constructed model, we affirmed that the bistability loops are caused by the two-mode competition and saturable absorptive effect in the two-section hybrid-coupled cavity.Bistable loops are observed from the L-I curves by the steadysolution of the constructed model, which are in accordance with the experimental results.In addition, we found that compared with the proportion of the cavity volume, the proportion of photon number confined in each cavity is more efficient for the control of the width of bistable loops in a wide range, which makes the proposed hybrid square-rectangular laser more particular than traditional two-section bistable lasers.Furthermore, all-optical flip-flop was simulated using a pair of optical trigger pulses at two competed mode wavelengths as set/reset signals.We notice that an optimal trigger energy is required in order to balance the transition time and turn-on oscillation on the transition edge.Under a strong nonlinear gain, minimum trigger energies of 14.2 fJ and 3.2 fJ are obtained for the switching of two competed modes at detuning frequencies of −10 GHz and −20 GHz, respectively.With low power consumption and simple fabrication processes, we expect that the tunable device can be densely integrated to fast-optical information processing systems.

FIG. 1 .
FIG. 1.(a) Schematic diagram of the HSRL, and an isolation trench is formed by ICP etching to ensure the electrical isolation between the gain and absorptive sections of the FP cavity and square microcavity.(b) Output power coupled into a single-mode fiber versus injection current I FP at thermoelectric cooler temperatures of 289, 293, 296, and 300 K. Lasing spectra at (c) 49, (d) 51, (e) 53, and (f) 55 mA with the increase in IFP and at (g) 49, (h) 51, (i) 53, and (j) 55 mA with the decrease in I FP at 289 K.

FIG. 2 .
FIG. 2. (a)Quasi-Fermi level separation energy versus carrier density N at 280, 300, and 320 K. Symbols are the results obtained from Eqs. (10)-(15) with the gain part and (16) with the absorptive part.Solid lines are the unified fitting curves of Quasi-Fermi level separation energy.Gain spectra with (b) the carrier density N as a parameter at 300 K and (c) the temperature T as a parameter at N = 1.6 × 10 18 cm −3 .(d) Gain (absorption) spectra with the temperature T at N = 1.0 × 10 18 cm −3 obtained by the constructed phenomenological gain model.

FIG. 3 .
FIG. 3. The output powers coupled into the fiber versus increased and decreased I FP as solid and dashed lines, respectively, at TEC temperatures of 280, 285, 290, 295, and 305 K.

FIG. 4 .FIG. 5 .
FIG. 4. (a) Output powers of M1 (red lines) and M2 (blue lines) versus I FP ; the solid and dashed lines represent the increase and decrease in I FP , respectively; (b) carrier density in the FP and the square cavity with the increase (solid lines) and decrease (dash lines) in I FP with temperature of 295 K.

FIG. 6 .
FIG. 6. Dynamic transition response at λ 1 and λ 2 of the device triggered by set/reset signals with a pulse width of 100 ps and pulse energies of 47 fJ and 13 fJ.and the normalized trigger pulse represented by the dashed lines.

FIG. 10 .
FIG. 10.(a) Rise and (b) fall edges of the mode switching from M1 to M2, triggered by optical trigger energies at M2 of 4, 6, and 9 fJ and (c) rise and (d) fall edges of the mode switching from M2 to M1 triggered by optical pulse energies at M1 of 27, 29, and 33 fJ, for the device with γ = 0.6.

TABLE I .
Parameters of the HSRL.