## Abstract

We investigate mode multistability, i.e. coexistence of direction bistability and wavelength bi/multistability in microring lasers (MRLs) theoretically and numerically. We derive the expressions for conditions required for mode multistable operation in microring lasers based on a nonlinear multimode model with nonlinear effects stemming from carrier density pulsation, carrier heating and spectral hole burning included. We find theoretically that lasing mode can be selected from the multistable modes by external optical injection through gain saturation, and removal of the external optical injection will not affect the stability of the established lasing mode. Numerical results on all-optical multistate flip-flop function demonstrate that switching between multistable modes can be induced by trigger signals with each states self-sustained after the removal of the trigger signals in a 50µm-radius microring laser.

© 2013 OSA

## 1. Introduction

All optical technology is very attractive because of its potential to overcome the electronic bottleneck limits for processing ultra-wideband and ultra-fast signals. Plenty of schemes have been proposed and investigated in these years for the implementation of the photonic counterparts of conventional electronic systems, such as the memory [1], logic [2] and switch [3]. Semiconductor disk or ring lasers, offering advantages of high speed, robustness, small size, low power consumption, have been investigated as an attractive candidate for this long-term dream [4–6]. Especially they have been realized on silicon platform recently for the purpose of optical sources and memory showing a great potential to integrate both optical and electronic functionalities on a common platform [7,8].

Recently, a unique feature of microring lases has been observed and experimentally demonstrated that stable unidirectional, single mode oscillation at set wavelength from both directions can be achieved without the requirement of any injection power to keep the laser locked in the desired mode, which is also known as mode multistability [9,10]. It shows a great future that a single photonics building block with multiple stable states capable of handling multiple wavelength information can be utilized to achieve lower power consumption, higher functional and integration densities.

Despite its interesting potential applications, the coexistence of directional bistability and wavelength bi/multistability in microring lasers has not been well understood. Mechanisms behind selection of lasing mode from multistable modes and switching between multistable modes are also still unclear. The direction bistable modes share the same wavelength but propagate in the opposite direction. Directional bistability in microring lasers is believed to be caused by mode competition between the counter propagating modes through nonlinear gain saturation which is described in a phenomenological way [11]. Wavelength bi/multistability in microring lasers refers to stable unidirectional, single mode oscillation at two or a set of wavelengths propagating in the same direction. Wavelength bi/multistability is investigated both analytically and numerically based on a nonlinear multimode model [12]. The results indicate that coupling and saturation between modes of different wavelengths through nonlinear gain leads to wavelength bi/multistability in microring lasers [12]. Later theoretical study on longitudinal mode multistability in ring and Fabry-Perot lasers [13] suggests that configuration is important for mode multistability, but the effects of nonlinear gain are not taken into account.

Actually, nonlinear effects are vital for understanding mode multistability which occurs as a result of mode competition in microring lasers. Resonant modes propagating in counter clock-wise (CCW) and clock-wise (CW) directions in microring cavity are travelling waves. Beating of modes at different cavity resonances in both directions modulates carrier density in the cavity at the beat frequency and leads to dynamic grating through nonlinear mechanisms. The dynamic grating results in a change of the gain of modes and coupling between modes both of which have great impact in mode competition.

In this paper, we investigate the mode multistability theoretically and numerically based on a time domain multi-mode nonlinear model with nonlinear effects from carrier density pulsation (CDP), carrier heating (CH) and spectral hole burning (SHB) considered. The theoretical and numerical methods used in this paper are similar to those used in studying wavelength bi/multistability in Ref [12]. The wavelength bi/multistable modes discussed in [12] have different wavelengths but propagate in the same direction. However, the mode multistable modes studied in this manuscript can either have different wavelengths but propagate in the same direction or share the same wavelength but propagate in different directions or have different wavelengths and propagate in different directions. We analyze the stability of potential multistable cavity modes and the condition required for mode multistable operation in microring lasers. Furthermore, we investigate the selection of lasing mode from multistable modes by external optical injection and the stability of the selected mode after the removal of injection theoretically. Numerical simulation of multistate all optical flip-flop function is then carried out to verify our theoretical results.

## 2. Theory

The dynamics of a multistable microring laser can be modeled by a couple of differential rate equations for the carrier density and the time evolution of the slowly varying envelop of the mode electric fields. The deviation of the carrier density from its threshold value is denoted by$\Delta N$:

*I*is the injection current,

*I*is the threshold current,

_{th}*e*for the elementary charge,

*V*for the volume of active region containing carriers,

_{c}*τ*for differential carrier lifetime,

_{N}*ε*for permittivity of free space,

_{0}*n*for group index,

_{g}*n*for the effective mode index,

_{0}*ω*for frequency of the linear gain peak,

_{pk}*v*for group velocity,

_{g}*g*for material gain of mode

_{a}*a*,

*E*for the complex slowly varying electric field of mode

_{a}*a*. Equation (1) takes into account the static carrier density change due to the intensity of modes, i.e., beating of the same modes.

The total electric field *E* in the laser is expressed as a sum of complex
field of both lasing and nonlasing modes in terms of *E _{a}*
as

*z*is the field propagation direction, with CCW as positive

*z*direction and CW as negative

*z*direction, and

*ω*accounts for the optical frequency of mode

_{a}*a*.

The dynamics of mode *a* is described by:

*Г*accounts for confinement factor,

*dg/dN*for differential gain coefficient,

*α*for linewidth enhancement factor associated with carrier density pulsation,

_{N}*τ*for photon lifetime, $\Delta {\omega}_{a}^{dis}$for total frequency shift of lasing mode frequency from the cold grid frequency. Equation (4) takes into account the effect of the carrier density beating due to beating of different modes and the nonlinear gain saturation effects which are described by the nonlinear susceptibility derived using density matrix method. ${\chi}_{bcd}^{3}$ for 3rd order material susceptibility including effects of CDP, CH and SHB for the interaction between modes

_{p}*b, c*and

*d*[14], with

where${\chi}_{bcd}^{x}$, *ε _{x}*,

*α*,

_{x}*τ*

_{x}_{, }${\eta}_{cd}^{X}$ are the nonlinear material susceptibility, nonlinear coefficient, linewidth enhancement factor, time constant, and diffusion coefficient associated with each mechanism.

*Ω = ω*is the beat frequency between modes

_{d}- ω_{c}*c*and

*d*.

*τ*is the dipole relaxation time. Gain

_{dp}*g*is related to

*χ*through $g=-\frac{\omega}{cn}\mathrm{Im}\left(\chi \right)=-\frac{\omega}{cn}\chi \text{'}\text{'}$.

*ζ*accounts for spatial overlap of longitudinal mode distributions of modes

_{abcd}*a, b, c*and

*d*, with

*ζ _{abcd}* would be very small for integration over many
wavelengths long unless resonant coupling condition i.e. ${k}_{a}-k{}_{b}+{k}_{c}-{k}_{d}=0$ is satisfied.

On the right hand side (RHS) of Eq. (4), the
first term accounts for the linear effect including amplitude gain and phase variations, and the
second term for the nonlinear coupling between modal fields. *F _{a}* is
the spontaneous emission Langevin noise term. Because effects from backscattering only weakly
couple the pair of counter-propagating modes with the same cavity resonance wavelength and do
not have noticeable effect on the mode multi-stability, they are not included in this model.

We refer to the lasing established mode which is closest to the material gain peak as mode
*L _{p}* (

*p = 1,2*) and one of the competing multistable modes as mode

*M*(

_{q}*q = 1,2*), where 1 and 2 represent the CCW and CW directions for wavelengths

*L*and

*M*respectively. We are interested in the dynamics and stability of the established mode and the competing multistable modes so we concentrate our theoretical analyses on the coupling between these modes. Assuming the laser is biased high above threshold current with quasi-single mode operation with mode

*L*as the dominant mode, we consider the situation illustrated in Fig. 1, where both wavelength

_{p}*L*and wavelength

*M*can propagate in either CCW or CW direction.

We approximate the linear material gain as parabolic around ${\omega}_{pk}$ with gain peak equals to${g}_{pk}$, and find the gain of each mode as:

*L*is assumed to be the one that is closest to the linear material gain peak with frequency ${\omega}_{L}={\omega}_{pk}+l\Delta \omega $ ($l=0,\pm 1,\mathrm{...}$) and wavelength ${\lambda}_{L}$, and the potential multistable

_{p}*M*is assumed to be farther away from the linear gain peak than mode

_{q}*L*with frequency ${\omega}_{M}={\omega}_{pk}+m\Delta \omega $ ($m=\pm 1,\pm 2,\mathrm{...}$), $\left|m\right|>\left|l\right|$ and wavelength ${\lambda}_{M}$where $\Delta \omega $ is the free spectral range (FSR). Because modes

_{p}*L*and

*M*have different wavelengths,

*1*and

*2*are used to represent two modes with the same wavelength but different propagation directions, i.e.

*1*for CCW and

*2*for CW direction of

*L*or

*M*respectively.

We find the following Eqs. for modes *L _{p}* and

*M*by evaluating Eq. (4):

_{q}If modes *L _{p}* and

*M*are co-propagating with

_{q}*p = q*, which is the case discussed in [12], the time evolutions of the normalized intensities of modes

*L*and

_{p}*M*are given by:

_{q}*p*and

*q*are removed for simplicity.

If modes *L _{p}* and

*M*are counter-propagating with

_{q}*p≠q*, nonlinear coupling between modes

*L*and

_{p}*M*through four wave mixing does not exist because phase matching condition is not satisfied. We find in this case:

_{q}*p*and

*q*are also removed for simplicity.

Consequently, if modes *L _{p}* and

*M*are counter-propagating, the time evolutions of the normalized intensities of

_{q}*L*and

_{p}*M*, which are derived based on Eqs. (11) and (12) and Eq. (14), possess the same form as Eq. (13).

_{q}Similarly, we find the time evolutions of the normalized intensities of modes with the same
wavelength in the CCW and CW directions (i.e. modes *L _{1}* and

*L*or modes

_{2}*M*and

_{1}*M*) by evaluating Eq. (4) as:

_{2}*L*and

*M*are removed for simplicity.

## 3. Analytical analysis and numerical results

#### 3.1 Mode multistability analysis

From the analyses and discussions above, we find that modes *L _{p}*
and

*M*degenerate into

_{q}*L*and

*M*since the time evolutions of their normalized intensities can all be described by Eq. (13) with whatever combination of

*p*and

*q*. Consequently, the notation

*p*and

*q*are removed in the following analyses for simplicity.

The stability of modes *L* and *M* described by Eq. (13) can be studied by phase plane analysis. As
shown in Fig. 2, two straight lines representing $d{S}_{L}/d{S}_{M}=0$ and $d{S}_{M}/d{S}_{L}=0$ intersect with axes *L* and *M* at
(*L _{l,m}*, 0) and (0,

*M*) respectively. Multistable operation of modes

_{l,m}*L*and

*M*can be achieved if

*L*>

_{l}*L*and

_{m}*M*<

_{l}*M*, i.e. the following conditions are satisfied:

_{m}Here each of modes *L* and *M* has a mode degeneracy of two
including CW and CCW directions as indicated by Eq.
(13), which means that two multistable states are of the same wavelength. The stability
of modes with the same wavelength but different propagation directions is analyzed in the same
way. As shown in Fig. 3, two straight lines representing $d{S}_{1}/d{S}_{2}=0$and $d{S}_{2}/d{S}_{1}=0$ intersect with axes *CCW* and *CW*
at (*CCW _{1,2}*, 0) and (0,

*CW*) respectively. We find that direction bistability can be achieved if

_{1,2}*CCW*and

_{1}>CCW_{2}*CW*, i.e. the following conditions have to be satisfied:

_{1}<CW_{2}Because modes considered in In Eqs. (18) and (19) propagate in different directions but share the same wavelength, it is obvious that In Eqs. (18) and (19) hold for any resonant mode in a normal ring laser.

Therefore, we find that the laser is mode multistable with at least four multistable modes if conditions described by In Eqs. (16) and (17) are satisfied. As indicated by In Eqs. (16) and (17), the coupling and saturation between resonant travelling modes through nonlinear mechanisms such as CDP, CH and SHB lead to mode multistability in semiconductor microring lasers.

It has been demonstrated by previous experiments [9,
10] that lasing mode of a multistable ring laser can be
controlled and selected by external optical injection. With external optical injections
included, the time evolution of the normalized intensity of modes *L* and
*M* are given by:

Where $\kappa $ is the injection coefficient, ${S}_{injL}$ and ${S}_{injM}$ are the external optical injection terms related to mode
*L* and mode *M* respectively.

We find that the lasing mode of a multistable microring laser can be selected from the
multistable modes with appropriate external optical injection through gain saturation as shown
in Fig. 4. As shown in Fig. 4(a), $d{S}_{L}/d{S}_{M}=0$is distorted by the external optical injection added to mode
*L*, flows are all attracted to the solution of *S _{M} =
0*, therefore mode

*L*is selected to be the lasing mode. Whereas Fig. 4(b) shows that $d{S}_{M}/d{S}_{L}=0$ is distorted by the external optical injection added to mode

*M*, flows are all attracted to the solution of

*S*, and therefore mode

_{L}= 0*M*is selected to be the lasing mode. Furthermore, because both modes

*L*and

*M*are multistable, the lasing mode will remain stable even after removal of the external optical injection signal as indicated by Fig. 2.

Similarly, the time evolution of the normalized intensity of modes with the same wavelength in the CCW and CW directions with external optical injections are given by:

*L*or mode

*M*respectively.

As shown in Fig. 5(a), with external optical injection added to the CCW direction of mode *L*
or *M*, $d{S}_{1}/d{S}_{2}=0$ is distorted and flows are all attracted to the solution of
*S _{2} = 0*, therefore CCW direction of mode

*L*or

*M*is selected to be the lasing direction. Whereas Fig. 5(b) shows that $d{S}_{2}/d{S}_{1}=0$ is distorted by the external optical injection added to the CW direction of mode

*L*or

*M*, flows are all attracted to the solution of

*S*, and therefore CW direction of mode

_{1}= 0*L*or

*M*is selected to be the lasing direction. The lasing direction will remain stable even after the removal of the injection signal since the microring laser is directional bistable as shown in Fig. 3.

#### 3.2 Numerical results

Numerical simulations are carried out based on a nonlinear multimode model including effects from CDP, CH and SHB. The model is described by Eqs. (1) and (4) with 30 modes from both CW and CCW directions considered. We assume that a multiple quantum-well material system is used to fabricate the device which is a ridge waveguide circular laser with a straight output waveguide coupled to the ring via a directional coupler. Main parameters for the device are given in Table 1.

We find that the microring laser can support ten different multistable modes by simulation,
i.e. modes *m _{0,CCW}* ~

*m*in CCW direction and modes

_{4,CCW}*m*~

_{0,CW}*m*in the CW direction where modes

_{4,CW}*m*and

_{0,CCW}*m*are the free running modes whereas

_{0,CW}*m*and

_{4,CCW}*m*are on the long wavelength side of the free running modes.

_{4,CW}Figure 6(a) shows the external trigger pulse signals with 10ns pulse width and 20ns time slot. Each trigger pulse signal representing one random number ranging from 0 to 9 will set the mode state to the corresponding multistable mode. We then carried out numerical study to verify the switching operations between different multistable modes by randomly selecting two zones (A and B) in the external injection pulse stream.

The mode states of the 10 selected input pulses in zone A appear as follows:
*m _{0,CCW}*,

*m*,

_{3,CCW}, m_{3,CW}*m*,

_{2,CW}, m_{4,CCW}*m*,

_{2,CCW}, m_{1,CCW}*m*, and

_{4,CW}, m_{1,CW}*m*(Fig. 6(b)), which cover all the supported multistable modes in the device. Triggered by the pulse stream, the oscillating mode switches from

_{0,CW}*m*in the CCW direction to modes

_{0,CCW}*m*,

_{3,CCW}, m_{3,CW}*m*,

_{2,CW}, m_{4,CCW}*m*,

_{2,CCW}, m_{1,CCW}*m*and

_{4,CW},*m*consecutively and finally switches to mode

_{1,CW}*m*in the CW direction. Each state is self-sustained which means that the selected multistable mode is stable after the removal of the trigger signal as shown in Fig. 7(a).

_{0,CW}The order for the injected trigger pulse stream in zone B is
*m _{0,CCW}*,

*m*,

_{0,CW}, m_{0,CCW}*m*,

_{4,CCW}, m_{2,CW}*m*,

_{4,CCW}, m_{4,CW}*m*,

_{2,CCW}, m_{1,CW}*m*,

_{3,CCW}, m_{1,CCW}*m*,

_{3,CW}, m_{2,CW}*m*and

_{3,CW},*m*and shown in Fig. 6(c). The combination includes not only switching between all the supported multistable modes, but also several set-reset operations between a few pairs of modes such as

_{1,CCW}*m*,

_{0,CCW}*m*and

_{0,CW},*m*. As shown by Fig. 7(b), the lasing mode is successfully switched to the selected states by the trigger pulse signals and self-sustained flip-flop operations are achieved after the removal of the signals.

_{0,CCW}Figure 8 shows the corresponding optical spectra of resonant modes in CCW (Fig. 8(a)) and CW (Fig. 8(b)) directions at different self-sustained states after the removal of trigger signals with reference to Fig. 7. It is clear that unidirectional single mode operation at the multistable mode is achieved as a result of an injection of a trigger signal to select the lasing mode.

## 4. Conclusions

We have theoretically and numerically investigated mode multistability consisting of direction bistability and wavelength bi/multistability in microring lasers. Dynamic grating generated by beating of cavity resonant modes from both directions leads to gain coupling and saturation between modes through nonlinear mechanisms such as CDP, CH and SHB. Mode multistability occurs due to strong nonlinear coupling between the competing modes. Expressions for conditions required for achieving mode multistability in microring lasers are derived from a nonlinear multimode model with nonlinear effects stemming from CDP, CH and SHB included. We also find theoretically that the lasing mode can be selected from the multistable modes by appropriate external optical injection through gain saturation, and removal of the external optical injection will not affect the stability of the established lasing mode. Numerical simulation on all-optical multistate flip-flop function aimed to verify the theoretical results demonstrates that external optical trigger signal can induce switching between multistable modes and therefore be used to select lasing mode from the multistable modes in a 50µm-radius microring laser. Both theoretical and numerical results are consistent with previous experimental findings and indicate that multistate microring lasers are very promising for future all optical scenarios and photonics integration technology.

## Acknowledgments

This work was sponsored in part by the National Natural Science Foundation of China under Grant 61107061, Grant 61107088, and Grant 61090393, Program for New Century Excellent Talents in University (NCET) under Grant NCET-12-0092, Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) under Grant 20100185120016, Project of international sci-tech cooperation and exchange research of Sichuan Province under Grant 2012HH0001, the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry 2012GJ002, the State Key Laboratory of Electronic Thin Films and Integrated Devices under Grant KFJJ201112, and State Key Laboratory on Integrated Optoelectronics under Grant 2011KFB008.

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