## Abstract

Both wavelength bistability (WB) and multiple bistability (MB) were predicted theoretically by Adams’ model in resonant optical amplifiers. We report here, for the first time to our knowledge, their experimental observation in 850nm Vertical-Cavity Semiconductor Optical Amplifiers (VCSOA). Clockwise hysteresis of WB is observed at constant input power while the input wavelength is swept across the gain window. MB is observed at a fixed operation wavelength biased on the long-wavelength side of the two separated Polarization-Dependent Gain (PDG) windows of the VCSOA by sweeping the optical input. The polarization of the input is set to a fixed angle with respect to the two intrinsic principal axes of the VCSOA. Two MB levels were experimentally observed at 160µW and 320µW, respectively. These observations are in good agreement with theoretical prediction by Adams’ model and may lead to multi-valued optical information manipulation.

©2007 Optical Society of America

## 1. Introduction

In the past decade, Vertical Cavity Semiconductor Amplifiers (VCSOAs) have drawn increasing research attention due to their intrinsic advantages over in-plane SOAs, such as high coupling efficiency, single-mode operation and low noise figure due to their small and circular cavity geometry [1–3]. The observation of optical Power Bistability (PB) in VCSOAs in 2002 [4], paved the way to various all-optical logic gates based on nonlinear VCSOAs including the recent demonstration of cascadable VCSOA inverters [5]. The small vertical structure and low power dissipation of VCSOAs makes them favorable for dense on-chip integration of optical logic arrays. In addition, compared with in-plane SOAs, VCSOA based logic elements typically exhibit lower switching power, and higher extinction ratio at logic output levels.

Although Adams’ model predicts that other types of bistabilities should be observable in VCSOA’s [6], it is for the first time to our knowledge that we report here, the observation of two different types of optical bistability in 850nm VCSOAs operated in reflection mode. Wavelength Bistability (WB), which has been previously reported in 1550nm VCSOAs, exhibits clock-wise hysteresis when the input wavelength is modulated while maintaining the input power constant [7]. This feature can be used for developing optically controlled wavelength filters and wavelength based logic operations in addition to developing a better understanding of non-linear behavior of VCSOAs. On the other hand, the observation of two bistable regions in the input-output power transfer characteristics that is Multiple Bistability (MB) may have various applications including multi-level logic operations, optical analog-digital conversion, polarization modulation/switching, waveform reshaping etc. MB is observed at a fixed wavelength at the long-wavelength side of the Polarization-Dependent Gain (PDG) windows of VCSOAs when the input polarization is linear and set at specific angle with respect to the two intrinsic principal polarization axes of VCSOAs.

In section 2, we review the physical mechanisms governing optical bistable behavior of VCSOAs using Adams’ model. Section 3 details the experimental observation of WB and its inherent relation to Power Bistability (PB). The operation principle, experimental and simulation results of MB are presented in section 4, followed by our conclusions.

## 2. Theory

The physical origin of optical bistabilities observed in a SOA mainly arises from the dependence of nonlinear refractive index on the carrier concentration in its Fabry-Perot (FP) cavity [6, 8]. The mechanism has been well described mathematically by the model proposed by Adams in the 1980’s [6]. Such a model is derived from two primary equations. The first is the classic transmission equation of the FP cavity, which is a function of the single-pass phase shift *ϕ* inside the cavity. The second is the nonlinear refractive index equation, where the impact of nonlinear refractive index is modified into the expression for *ϕ* [6, 8, 9]. Based on Adams’ model, the calculated Transmissitivity/Reflectivity of FP SOA exhibits multiple states for the same value of the single-pass phase shift *ϕ* within its bistable region. Optical bistable switching occurs as soon as the single-pass phase shift *ϕ* crosses its bistable threshold ϕ_{TH} [6, 10]. The full Adams’ model for SOAs operated in reflection mode is described as follows. First, Eq. (1) and Eq. (2) shown below relate the impact of intra-cavity photon intensity and initial phase detuning *ϕ*
_{0} on *ϕ* and net gain *g* at steady state, respectively.

where *L*, *Γ*, *g*
_{0}, *b*, *α* are the effective cavity length, confinement factor, unsaturated net gain per unit length, the linewidth enhancement factor and the effective loss coefficient, respectively. The initial phase detuning *ϕ*
_{0} is defined by 2*πL*/*λ*, where *λ* is the input wavelength. The saturation intensity **I*** _{s}* is determined by

*E*/

*Γ*

*aτ*, where

*E*is the photon energy,

*a*is differential gain and

*τ*is the electron lifetime. The intracavity average intensity

**I***is given by*

_{av}where *R*
_{1} and *R*
_{2} are the reflectivities of the front and rear cavity mirrors, respectively. P_{x} is the input scaling parameter used to convert input power *P*
* _{in}* to the input intensity. The actual value of P

_{x}is determined by the aperture size of the SOA and coupling loss. The output power of an SOA in reflection mode has the form of

where P_{y}, is output scaling factor performing the similar functions as P_{x}. Based on Eqs. (1)–(4), the different types of optical bistabilities in an SOA can be predicted [6,10].

## 3. Wavelength bistability

It is noted in Eqs. (1)–(3) that the single-pass phase shift *ϕ* is actually dependent on both input power *P*
* _{in}* and input wavelength

*λ*. Thus, mathematically, the bistable point (

*λ**,

*P*

**) corresponding to a certain phase bistable threshold*

_{in}*ϕ*

*can be calculated based on Adams’ model.*

_{TH}Accordingly, the same *ϕ*
* _{TH}* can be probed by either sweeping

*P*

*with a fixed*

_{in}*λ** (

*power bistability*) or sweeping

*λ*with a fixed

*P*

** (*

_{in}*wavelength bistability*).

The WB measured in a proton-confined VCSOA manufactured by Emcore with I_{bias}=0.95I_{th} is shown in Fig. 1. In such a measurement, the input polarization must be precisely aligned to one of two separated intrinsic PDG windows that typically result from material birefringence [11]. It is clearly shown in Fig. 1 that, with increasing input power *P*
* _{in}*, the wavelength-gain characteristics show increasingly enhanced clockwise hysterisis. In addition, the increase in input power also results in the widening of the bistable region and a red-shift of the gain hysteresis, which is consistent with the prediction of Adams’ model [6].

To prove that WB and PB are inter-related through a bistable operating point (*λ**, *P*
* _{in}**), a special characterization experiment was conducted. Two normalized gain windows for P

_{in}=100nW and P

_{in}=30µW are measured and plotted in Fig. 2 by sweeping the input wavelength. Note that the plot for P

_{in}=30µW shows two bistable threshold wavelengths

*λ*

_{1}(841.726nm) and

*λ*

_{2}(841.735nm) corresponding to two bistable points (

*λ*

_{1},

*P*

*=30µW) and (*

_{in}*λ*

_{2},

*P*

*=30µW). In addition, two power bistable transfer characteristics are measured and plotted with their operating wavelengths kept at*

_{in}*λ*

_{1}and

*λ*

_{2}, respectively. The plots in Fig. 3 clearly show a power bistable threshold of ~30µW that is shared by both power bistable curves. Therefore, two bistable points measured on the same wavelength bistable characteristics can also be observed at their corresponding power bistable characteristics, respectively, as expected from Adams’ model as discussed above. Conversely, it is expected that two bistable points occurring on a power bistable transfer curve can also be observed at two corresponding wavelength bistable transfer curves; where each one of the bistable points is observable on one of the corresponding wavelength bistable curve.

## 4. Multiple bistability

Multiple bistability is defined to occur when two or more bistable regions can co-exist in the input-output transfer characteristics. MB is usually characterized by multiple logic levels that occur for different input bistability thresholds. MB has been observed and studied previously in a FP type Laser Diode amplifier, with a cavity length (1.35mm) more than a thousand times larger than its input wavelength (1.3µm). Such a large cavity length allows successive input-induced effective longitudinal mode switching, resulting in MB in laser diode amplifiers [12]. Such an explanation, however, does not apply to VCSOA MB because of their short cavity in the order of the wavelength exhibiting a single-longitudinal mode. Here, we propose and demonstrate a novel explanation of MB observed in VCSOA.

As mentioned above, owing to material birefringence, VCSOAs exhibits two split PDG windows [11]. Figure 4 shows the two intrinsic PDG windows of a VCSOA, which are probed by a 100nW input when the device is biased at 95% of its threshold. To obtain the MB, the wavelength of the optical input is set at the long-wavelength side of both PDG windows, as shown in Fig. 4, while the linear polarization of the input is adjusted to have a fixed angle θ with respect to the intrinsic polarization direction of the PDG window with the shorter peak wavelength.

In such an operating scheme, the input power can be decomposed into P and S directions in a simple form given by

Because of the finite separation of the two PDG windows (20pm in this case), the two decomposed beams P_{in-S} and P_{in-P} have different wavelength detunings (D for *P*
_{in-S} and D+20pm for *P*
_{in-P}), resulting in two different bistability threshold powers and different bistable amplitudes [5, 9, 10]. Accordingly, the superposition of these two bistabilities in the P and S direction leads to multiple bistable levels for appropriately adjusted detuning and polarization angle θ.

In Fig. 5 we show under the operating conditions D=7pm, θ=20o a multiple bistable transfer curve (see Fig. 5). This measured multiple bistable curve clearly exhibits two bistable regions with bistable output power levels of 160µW and 320µW, respectively. To gain further insight in the interactions of the two input polarization components, the total output power of the VCSOA is decomposed into P and S directions and monitored simultaneously by adding a Polarization Beam-splitter (PBS) at the output. The PBS polarization orientation is aligned with that of VCSOA intrinsic polarization direction. The experimental results shown in Fig. 5 indicate that a polarization switching from P to S occurs at the second bistable switching threshold power around 25µW. This polarization switching can be understood as a result of the Cross-Gain Modulation (XGM) effect between *P*
_{in-S} and *P*
_{in-P} [13]. Specifically, as the output in the P direction is switched to a high level via its power bistability, the carrier density of the VCSOA is consumed mainly by the strong stimulated emission induced by P_{in-P}. Such a decrease in the carrier density suppresses the amplification gain *G* (*G*=*P*
* _{out}*/

*P*

*) in the S direction by saturating the material gain for*

_{in}*P*

_{in-S}and shifting its resonant frequency to a different value. Thus, a bistable polarization switching is observed when the strong bistable switching in the P direction imprints its sharp and bistable transition inversely on the output in S direction.

An investigation of this polarization response of the VCSOA to various input polarization angles θ was conducted and results are shown in Fig. 6 and Fig. 7, where the transfer characteristics in the P and in the S direction are shown in Fig. 6, and in Fig. 7, respectively. We found that the polarization response is very sensitive to the input polarization angle. For VCSOA devices we tested a strong polarization switching was observed only when the input polarization angle θ was set carefully to 20 degrees.

To explain and verify the above observations, a simulation was conducted based on a modified Adams’ model. In the modified model, the optical input *P*
* _{in}* is first decomposed into

*P*

_{in-S}and

*P*

_{in-P}based on Eqs. (6) and (7). Then, two sets of Adams’ Eqs. (1), (3) and (4) are used to describe the bistable behavior in S and P directions, separately, and two sets of parameters

*P*

*,*

_{out}*I*

*,*

_{av}*L*and

*ϕ*are assigned for each direction. The material birefringence in a VCSOA can be modeled by setting different values for

*L*

*and*

_{S}*L*

*, where the difference between them is proportional to the wavelength separation of the intrinsic PDG windows. Since*

_{P}*P*

_{in-S}and

*P*

_{in-P}share the carriers in the same VCSOA cavity, Eq. (2) is still valid and applicable to both beams. Therefore,

*I*

*in Eq. (2) can be expressed as*

_{av}The modification is used essentially to couple two sets of Adams’ equations via Eq. (2). Based on such a modified Adams’ model, the multiple bistable transfer characteristics and polarization switching are simulated. The simulation results shown in Fig. (8) are obtained under operation conditions of I_{bias}=0.95I_{th}, θ=10° and D=16pm, where D is the detuning of input beam relative to the peak of intrinsic polarization-dependent gain window at longwavelength side (other parameters were chosen based on values reported in [4, 10]). As shown in Fig. (8), the phenomena of both MB and polarization switching of the VCSOA observed experimentally can be predicted by the modified Adams’ model. The simulation results are in good agreement with the experimental observation demonstrating the validity of our proposed explanations.

## 5. Conclusion

In this paper, we have reported, for the first time to our knowledge, the observation of both wavelength and multiple bistabilities in an 850nm VCSOA. In addition to demonstrating WB, we also examined the connection between PB and WB experimentally in a VCSOA for the first time. The experimental results are in good agreement with our theoretical prediction that the same bistable point (*λ**, *P*
* _{in}* *) is shared by both PB and WB. Furthermore, an original operating scheme for realizing MB in a VCSOA has been proposed and verified experimentally. The experimental results show both MB and polarization switching, and these phenomena are well predicted by using a model based on the modification of Adams’ model. We believe our observations and modeling will help engineering and optimization of all optical nonlinear logic operations based on resonant optical amplifiers such as VCSOAs.

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