SOA Model and Design Guidelines in Lossless Photonic Subsystem

Abstract

We propose a new practical analytical model to calculate the performance of amplitude-modulated systems, including semiconductor optical amplifiers (SOA). Lower and upper-performance bounds are given in terms of signal quality factor (Q) concerning the input signal pattern. The target is to provide a design tool for gain elements included in photonic integrated circuits (PIC) to compensate for their insertion loss. This subject is a critical issue, for example, in the arrays of optical transmitters with silicon photonics modulators used for interconnection applications. Due to implementation limitations, the design of an SOA embedded in a PIC is considerably different with respect to the use of SOAs as line amplifiers in optical networks. SOA amplified spontaneous emission (ASE) and gain saturation effects have been included in the model, together with the input signal extinction ratio and the receiver electrical filter. Each degradation effect provides its own contribution to the signal integrity in terms of signal-to-noise ratio (SNR) or inter-symbol interference (ISI). The model shows that the SOA operation at low extinction ratios, typical in optical interconnect applications, is substantially different from the operation at higher extinction ratios used in transport networks. The model is validated through numerical simulations and experiments. Finally, two examples are provided for dimensioning a PIC system and optimizing the SOA parameters.

1. Introduction

The growing demand for fast, miniaturized, and power-efficient optical devices leads to provide growth opportunities for photonic integrated circuits (PICs) to process or distribute information. Passive PICs are widely utilized, for example, for optical beam steering [1], integrated photonic filters [2], and integrated silicon photonics transceiver [3] nevertheless, active PICs have numerous applications in optical systems such as lossless reconfigurable optical add-drop multiplexers (ROADM) in the fronthaul network [4]. The PICs allow optical systems to be more compact than discrete optical components. However, some PIC components, such as modulators, multiplexers, and splitters, cause a high loss. In addition, in a PIC, active and passive optical components are interconnected by lossy optical waveguides and need to be coupled with input and output fibers, which adds even more attenuation. One of the most promising solutions for compensating PIC losses is to integrate a semiconductor optical amplifier (SOA) on-chip in combination with other passive components. Numerous research studies have been reported on SOA technology and applications, both as on-chip or stand-alone components [5, 6]. SOAs can be designed to amplify signals in a very wide band range, over 100 nm optical bandwidth [7], to implement loss-less high-speed systems [8]. They are available in both C-band and O-band. When designed in O-band, they are commonly used in coarse wavelength-division multiplexing (CWDM) systems [9]. Manifold researches have shown their role in optical communication networks. Schmuck et al. [10] presented SOA performance in open metro-access to create a transparent and reconfigurable optical ring network; in [11], they reported the importance of SOA efficiency in PON-applications as a booster and pre-amplifier. Ramírez et al. [12] described the essential role of SOA to extend the link power budget in short reach networks in an 8 × 50 Gb/s WDM system. SOAs are also considered as a substantial component for ultrafast all-optical signal processing devices, as wavelength converters [13], all-optical gates [14], and optical de-multiplexers [15]. Recently, on-chip SOAs attracted a lot of attention due to their large-scale integration capability, low cost, and offer of more compact and smaller devices with low losses. Over the last few years, SOA integration technology has significantly been developed. Different technologies have been demonstrated for SOAs use in photonic chips, either by hybrid or by monolithic integration solution. Developing SOA is challenging on both pure InP platforms and hybrid III-V-on-silicon platforms. Specific coupling/bonding techniques are required such as flip-chip bonding [16, 17], die-to-wafer bonding [18], edge coupling [19], spot size converter (SSC) [20] and chip-to-chip butt coupling [21]. Furthermore, SOA monolithically integrated with photodiodes has been proposed as a promising technique for low-cost, high-speed, high-sensitivity SOA_PIN receivers [22, 23]. The recent demonstration of SOA heterogeneous integration on a silicon substrate by direct bonding of InP-based active region to the substrate introduced new possibilities for advanced PICs in the high wavelength areas [24]. More recently, a U-bend design has been proposed for III-V gain devices, such as SOAs, for simplifying the butt-coupling between the III-V chip and silicon-on-insulator photonic circuit [25]. All these techniques have been proposed to enable the SOA implementation on-chip for different applications. SOAs can operate either in the linear regime, as reach-extender amplifiers in gigabit passive optical networks (GPON) [26], or in the nonlinear regime, as an all-optical switching [27, 28]. SOA-based chip-level applications are not limited to optical communication systems but also include medical sensors, industrial and environmental monitoring [29], and nonlinear optics [30]. Depending on application requirements and chip fabrication constraints, the SOA working operation such as design may be very different. One of the most significant parameters is the gain value. SOAs can be designed to operate at high gain, as required on the receiver side to increase sensitivity [11], as well as at low gain/high output saturation power, as needed at the transmitter side in order to increase the lunch power [31]. Generally, chip-scale tunable lasers utilize a booster SOA. In integrated microwave photonics, a high gain SOA is required in combination with a tunable comb laser [32] or tunable ring-resonator-based lasers [33, 34]. In Ref. [35], a vertical-cavity surface-emitting laser (VCSEL) is co-packaged with a high gain SOA for OCT and LiDAR applications. Device parameters (e.g., active region dimensions and gain) and system parameters (e.g., input signal extinction ratio (ER)) have both a strong influence on SOA operation. Directly modulated lasers (DMLs), typically used in low-cost transmission systems, have limited extinction ratio, which affects SOA dynamics [36] and decreases the signal quality. To solve this issue, the external light injection has been suggested by Lu et al. [37] for a 10 Gb/s optical transport system based on VCSELs and high gain SOAs. They used this technique to increase the SOA saturation power. By the increment of the injected light, they kept the SOA in the linear operation regime. In this paper, we demonstrate the possibility of using a low gain SOA after a low ERtransmitter, such as DMLs, without any additional component. SOAs have also been indicated for compensating on-chip losses [38]. Utilizing SOA with high gain for compensating PIC’s component losses has already been proposed for up to 164 optical components on a fully integrated free-space beam steering chip [39]. In the above-mentioned SOA-based PICs the quality factor of the amplified signal is not estimated. Assessing the quality of the optical signal is of paramount importance in the design of an optical system. Knowing the SOA impact on the signal quality during the design phase can remarkably reduce the number of fabrication runs, leading to significant cost savings. For access network applications, the SOA influence on the Q-factor has been theoretically and experimentally evaluated for finding the input power dynamic range (IPDR) as a function of bitrate, wavelength, and bias current [40]. That model is based on an empirical tolerance factor, a lower limit of the IPDR due to a low OSNR, and an upper limit due to gain saturation.

In this chapter, we propose a new analytical model for the SOA impact on the signal quality factor by taking into account the effects of gain compression, ASE noise, signal extinction ratio (ER), and receiver filter shape and bandwidth. The model assumes single-channel signal transmission, with conventional on-off key (OOK) non-return-to-zero (NRZ) modulation. The receiver is assumed to sample the signal at half of the bit time, as in all practical implementations. The proposed model provides two analytical equations for the Q-factor in the worst and best system operating conditions. For the analytical model proposed in this investigation, we considered some SOA specifications, such as active region dimensions and several effects that influence SOA performance, intending to estimate the Q-factor of an SOA performance before its actual integration into the system. The model can be used as a guideline to design a proper SOA in PIC or as a stand-alone device. The proposed model provides a tool for optimizing the SOA design. We apply the model considering two specific examples of SOA integrated into photonics interconnection systems or as a stand-alone device. The numerical model we used is essentially similar to that described in [41]. The steady-state and dynamic analyses are performed numerically for the SOA design using a parabolic gain model. The dynamic model used to simulate the SOA behavior is based on the numerical solution of the rate equation for carrier density, for a multi-section SOA. To solve the rate equations of each section, a standard ordinary differential equation (ODE) is used. The study is here limited to intensity variation neglecting the chirp effect that may be included in a more complete description when complex modulation formats are used in the system as well as long fiber span. We investigated the design of a low-gain SOA device that can be utilized as an on-chip loss compensator, or as a booster for a DML, through the analytical model and numerical simulations for various operating conditions. We estimate performance in terms of quality factors. Finally, the model is experimentally validated through a 10 Gb/s OOK WDM transmission system including a commercial SOA. We further discuss the extension of the model to higher-order modulation formats. In the following sections of the chapter, the proposed model is described in detail. In Section 2, the model is introduced together with its main parameters. Section 3 is divided into three subsections where gain compression, extinction ratio, and receiver filtering effects are modeled, respectively. Experimental setup and validation are described in Section 4. Section 5 shows two design examples and possible model extensions. Finally, conclusions are summarized in Section 6.

2. Model

The system under investigation is described through the block diagram in Figure 1a.

Figure 1.

(a) Block diagram, and (b) System representation. OF: optical filter, EF: electrical filter, and STG: saturated gain.

An optical transmitter generates the signal $P_{\text{in}}\left(t\right)$at the wavelength λas a sequence of binary symbols:

$$P_{\text{in}}\left(t\right)=\sum _{n=-\infty }^{+\infty }{b}_{n}p\left(t-n{T}_b\right)$$E1

where $b_n\in \left[P_{\text{in}}^L,P_{\text{in}}^H\right]$are the transmitted bit power [mW], and p(t) is an ideal rectangular pulse whose time duration is equal to T_b. When the probability of ones and zeros is equal, the average input power is $\overline{P_{\text{in}}}=\left(P_{\text{in}}^H+P_{\text{in}}^L\right)/2$, and the extinction ratio is defined as $\mathit{ER}\stackrel{\Delta }{=}{P}_{\text{in}}^H/P_{\text{in}}^L$. The signal is amplified through an SOA that provides a gain, $G\left(t\right)$, depending on the SOA physical parameters, such as the driving current $I_{\mathrm{dc}}$, and the input signal.

$$P_{\text{out}}^{\mathrm{SOA}}\left(t\right)=P_{\text{in}}\left(t\right)G\left(t\right)$$E2

In the hypothesis of λcorresponding to the SOA gain peak, the saturated SOA gain must satisfy the equation [26]:

$$G\left(t\right)=G_{0}exp\left(-\frac{G\left(t\right)-1}{G\left(t\right)}\frac{P_{\text{out}}^{\mathrm{SOA}}\left(t\right)}{P_{\mathrm{sat}}}\right)$$E3

being G₀ the unsaturated gain (small-signal gain) and $P_{\mathrm{sat}}$the saturation power. At the same time, SOA generates noise n(t), modeled as a white Gaussian noise, whose power spectral density is a function of the gain and the population-inversion factor n_sp(t), such as Eq. (4).

$$N_{\mathrm{ASE}}\left(t\right)=2n_{\mathit{sp}}\left(t\right)\frac{\mathit{hc}}{\lambda }\left[G\left(t\right)-1\right]$$E4

where his the Plank constant and c isthe speed of light in the vacuum. After amplification, in order to reduce the impact of the noise, an optical band pass filter is included with response time h_o(t) and 3-dB bandwidth B_oso that

$$P_{\text{out}}^F\left(t\right)=P_{\text{out}}^{\mathrm{SOA}}\ast h_o\left(t\right)\text{and}{P}_{\mathrm{ASE}}\left(t\right)=N_{\mathrm{ASE}}\left(t\right)\ast h_o\left(t\right)$$E5

By neglecting thermal and quantum noise, the photodetected current is:

$$I\left(t\right)=R{\left|\sqrt{P_{\text{out}}^F\left(t\right)}+\sqrt{P_{\mathrm{ASE}}\left(t\right)}\right|}^2=R\left[P_{\text{out}}^F\left(t\right)+2\sqrt{P_{\text{out}}^F\left(t\right)P_{\mathrm{ASE}}\left(t\right)}+P_{\mathrm{ASE}}\left(t\right)\right]$$E6

Being R[A/W] the photodetector responsivity. The first term in Eq. (6), represents the signal, whereas both the other two terms are noise contributions. At the receiver input, an electrical filter with bandwidth B_eis considered, giving a bandlimited new signal including the additive noise

$$I^F\left(t\right)=I\left(t\right)\ast h_e\left(t\right)$$E7

where h_e(t) represents the electrical filter impulse response. It is usual to evaluate the performance of an optical communication system at the optimal sampling time. Here, for the sake of simplicity and for considering a realistic clock-data recovery circuit, a fixed sampling time at T_b/2 will be considered, so that the output samples take the values

$$I_{\text{out}}^k=I^F\left(k{T}_b/2\right)$$E8

Each sample represents a distorted binary symbol with an additional white Gaussian noise whose variance depends on the instantaneous noise contribution as described in Eq. (6). When an OOK bit stream as represented in Figure 2a is propagated through the system, the photo detected current will assume the generic behavior of Figure 2b including those distortions that can be ascribed to the saturated gain and the additional noise. By neglecting the effect optical and electrical filters, it is possible to identify the highest and lowest values for samples corresponding to each input bit-level providing:

Figure 2.

(a) Input signal example, (b) distorted output. Sampled values are marked at T_b/2 interval and they are shown with circles.

$$I_b^L=min\left\{I_{\text{out}}^k\right\},I_w^L=max\left\{I_{\text{out}}^k\right\}$$E9

when $P_{\text{in}}^L$is transmitted, and

$$I_b^H=max\left\{I_{\text{out}}^k\right\},I_w^H=min\left\{I_{\text{out}}^k\right\}$$E10

when $P_{\text{in}}^H$is transmitted.

In the simple case of linear gain, $I_b^L=I_w^L=I_L$and $I_b^H=I_w^H=I_H$. For what concerns noise contributions to the same samples (at T_b/2), we assume a white Gaussian noise whose variances, associated with the input bit $P_{\text{in}}^L$and $P_{\text{in}}^H$, can be written as:

$${\sigma }_{L,H}^k=R\left\{\left[2P_{\text{out}}^F\left(k{T}_b/2\right)P_{\mathrm{ASE}}\left(k{T}_b/2\right)\right.\right.{\left.\left.+P_{\mathrm{ASE}}^2\left(k{T}_b/2\right)\right]B_e/B_o\right\}}^{1/2}$$E11

Noise variances assume four discrete values in correspondence of best and worst cases for low and high input power that will be indicated with ${\sigma }_{L,H}^{b,w}$.

As it is shown in Figure 3, when the optimal threshold is used for bit decision [42] and bit one and zeros have the same probability, the bit error rate (BER) can be found, in the two cases, by the highlighted areas so that:

Figure 3.

Probability density function$P_{0,1}\left(I_b\right)$corresponding to the best case at mean one$I_b^H$and zero$I_b^L$levels (with a standard deviation${\sigma }_H^b$and${\sigma }_L^b$) with the optimal threshold for bit decision$I_b^{\mathit{Th}}$in red and probability density function of the worst$P_{0,1}\left(I_w\right)$at mean one$I_w^H$, and zero$I_w^L$levels (with${\sigma }_H^w$and${\sigma }_L^w$) with the optimal threshold for bit decision$I_w^{\mathit{Th}}$in light blue. The hatched red area determines the BER_b for the best case while the hatched blue area represents the BER_w for the worst case.

$${\mathrm{BER}}_w=\frac{1}{2}\frac{1}{{\sigma }_L^w\sqrt{2\pi }}{\int }_{I_w^{\mathit{th}}}^{\infty }exp\left[-\frac{1}{2}{\left(\frac{I-I_w^L}{{\sigma }_L^w}\right)}^2\right]\mathit{dI}+\frac{1}{2}\frac{1}{{\sigma }_H^w\sqrt{2\pi }}{\int }_{-\infty }^{I_w^{\mathit{th}}}exp\left[-\frac{1}{2}{\left(\frac{I-I_w^H}{{\sigma }_H^w}\right)}^2\right]\mathit{dI}$$E12

$${\mathit{BER}}_b=\frac{1}{2}\frac{1}{{\sigma }_L^b\sqrt{2\pi }}{\int }_{I_b^{\mathit{th}}}^{\infty }exp\left[-\frac{1}{2}{\left(\frac{I-I_b^L}{{\sigma }_L^b}\right)}^2\right]\mathit{dI}+\frac{1}{2}\frac{1}{{\sigma }_H^b\sqrt{2\pi }}{\int }_{-\infty }^{I_b^{\mathit{th}}}exp\left[-\frac{1}{2}{\left(\frac{I-I_b^H}{{\sigma }_H^b}\right)}^2\right]\mathit{dI}$$E13

When threshold is optimized,

$$\mathrm{BE}{\mathrm{R}}_w=\frac{1}{2}\text{ercf}\left(\frac{Q_w}{\sqrt{2}}\right)\text{and}\mathrm{BE}{\mathrm{R}}_b=\frac{1}{2}\text{ercf}\left(\frac{Q_b}{\sqrt{2}}\right)$$E14

where,

$$Q_w=\frac{I_w^H-I_w^L}{{\sigma }_H^w+{\sigma }_L^w}\text{and}{Q}_b=\frac{I_b^H-I_b^L}{{\sigma }_H^b+{\sigma }_L^b}$$E15

In the linear case,

$$Q_w=Q_b=Q=\frac{I_H-I_L}{{\sigma }_H+{\sigma }_L}$$E16

In the simulation and experimental results, the quality factor (eye-opening) is considered based on the signal received by the decision circuit that samples at the decision instant (T_b/2) as described in Eq. (8). The sampled values fluctuate from bit to bit around an average value I^Hor I^L, depending on the bit logical value in the bit stream [42]. The decision circuit compares the sampled value with a threshold value Th. In the analytical model, we estimate a boundary condition for the SOA quality factor based on the highest and lowest sampled values corresponding to each input bit-level. The sampled values located around an average value I_b^H /I_b^Lor I_w^H/I_w^Lprovide boundary conditions $Q_b$and $Q_w$, respectively, as described in Eqs. (9) and (10). The best and worst quality factors correspond to linear and nonlinear SOA operation as it will be clarified hereafter.

Figure 4 shows, as an example, 10 Gb/s NRZ signal eye diagrams when passing an SOA operating in nonlinear (top) and linear(bottom) regimes. On the left-hand side, analytical probability density functions (pdf) are reported in the two boundary cases; to the right, simulated pdfs are shown for comparison. SOA parameters are given in Section 4. When the input signal power is high enough (–12dBm in the example), SOA suffers for nonlinearities and the analytical model estimates a Q-factor to be $Q_w=13.3\mathrm{dB}$. By lowering the power down to –19 dBm, SOA is almost linear and the model returns $Q_b=12.2\mathrm{dB}$. For both cases, the extinction ratio of the rectangular input signal is 6 dB. In the same operating conditions, simulated values of Q-factor, as shown in Figure 4 right, are 13.8 and 11.9 dB for the nonlinear and linear cases, respectively, which are in good agreement with the analytical ones. The pdfs on the left, with the sampling time at the beginning of the bit, show the separated high and low levels for best and worst cases to assess the analytical $Q_b$and $Q_w$with the optimal thresholds (I_b^Th/I_w^Th). Experimental validation is provided in Section 4. When the optical input power exceeds the linear regime, the two bound values strongly depend on several parameters, including SOA gain saturation and recovery time, as well as input signal ER, bit pattern and rate, filters bandwidth, and shape.

Figure 4.

Normalized eye diagrams and probability density functions (pdf) for SOA operation in nonlinear (a), and linear (b) regimes, left side: shows pdfs considering$Q_b$and$Q_w$, and right side: shows pdfs considering actual Q-factor.

Although the model assumes single-channel transmission, it is even valid for multichannel systems. In fact, if the linear operation description is trivial, in the nonlinear case, the worst estimation is given when all the WDM channels are assumed to be synchronous with the same data stream. Since this analytical model is based on the gain variation in the SOA, any effect that has an impact on that, including nonlinear gain modulation, contributes to Q-factor degradation. For instance, consider the SOA performance in the WDM system (as in line amplifier) for 12 ×25 Gb/s NRZ transmission with 200 GHz channel spacing with the power per channel equal to –12 dBm and extinction ratio of 15 dB. In this case, the total power is –1.2 dBm (corresponding to –12 dBm per channel). Using the model for a single channel with –1.2 dBm, provides the estimated boundary conditions of the quality factor for all 12 channels. In the following different impairments will be studied and analytical expressions provided.

3. Performance estimation

3.1 Saturated gain

In order to isolate the effect of saturated gain, the following assumptions have been made:

1. $P_{\text{in}}^L=0\Rightarrow \mathrm{ER}=\infty \Rightarrow I_b^L=I_w^L=I_L=0$,

2. Ideal optical and electrical filters so that B_e/B_o=1.

When a binary pattern with an infinite extinction ratio is propagated through an SOA, each bit one is instantaneously amplified with a gain that depends on the previous bit sequences and can vary between the small-signal linear gain G₀ and the minimum value represented by the saturated gain G_sobtained for a constant input power $P_{\text{in}}^H$. The former case occurs when the pattern preceding bit one is composed of a number of consecutive zeros large enough for a complete gain recovery. This number depends on the carrier lifetime ${\tau }_c$and the bit time T_bas it is shown in [43]. The highest power at the sampling instant is obtained when τ_c≫ T_band can be approximated with $P_{\text{in}}^H{G}_0$. Where for the ($\frac{{\tau }_c}{{\mathrm{T}}_b}\ge 8$), the input power is considered as its average value [43]. On the other hand, the lowest output power is obtained for steady-state operation when SOA reaches saturation and can be expressed by $P_{\text{in}}^H{G}_s$. SOA gain recovery depends on the relation between ${\tau }_c$and $T_b$. Figure 5 clarifies the SOA gain dynamic operation in the case of constant input power (–12 dBm) for different $\frac{{\tau }_c}{T_b}$ratio values.

Figure 5.

Simulated eye diagrams (Power (mW)) for various bit rates at 1555 nm atI_dc = 70 mA, and at the constant input power (–12 dBm). Eye diagrams show the gradual changes in the Q-factor boundaries due to the ratio (${\tau }_c/T_b$).

When the bit rate is low, $T_b$is comparable with ${\tau }_c$and SOA gain can recover to its highest value (small-signal gain) within a few bit times when bit is zero. Each bit one, following one or more-bit zeros, experiences a gain that is high at the beginning of the bit time but it quickly decreases to the saturated value. Saturation depends on input power as well as bit distribution and modulation extinction ratio. As the bit rate increases, both gain saturation and recovery gradually flatten and SOA becomes transparent to the bit pattern if that does not include very long sequences of bit zero. The eye diagrams in Figure 5 do not include ASE noise.

Boundary conditions can be easily determined from Eq. (15), such as:

$$Q_w=\frac{I_w^H}{{\sigma }_H^w+{\sigma }_L^w}=\frac{{\mathrm{P}}_{\text{in}}^H{G}_s}{P_{\mathrm{ASE}}^0+\sqrt{{\left(P_{\mathrm{ASE}}^s\right)}^2+2P_{\mathrm{ASE}}^s{P}_{\text{in}}^H{G}_s}}$$E17

and,

$$Q_b=\frac{I_b^H}{{\sigma }_H^b+{\sigma }_L^b}=\frac{{\mathrm{P}}_{\text{in}}^H{G}_0}{P_{\mathrm{ASE}}^s+\sqrt{{\left(P_{\mathrm{ASE}}^0\right)}^2+2P_{\mathrm{ASE}}^0{P}_{\text{in}}^H{G}_0}},$$E18

where $P_{\mathrm{ASE}}^s$and $P_{\mathrm{ASE}}^0$are the ASE powers calculated through Eq. (4), in the saturated and linear conditions respectively. Figure 6a shows an ideal rectangular NRZ_OOK bit pattern with infinite ERlaunched through the SOA. The behavior of the detected signal pattern by neglecting SOA_ASE noise is shown in Figure 6b. While Figure 6c shows the effect of the SOA_ASE noise on the signal pattern as demonstrated in Eq. (6). The sampled values are shown with the dotted lines and the $I_b^H$and $I_w^H$marked by dash-dotted and dashed lines, respectively. In this pattern stream, the highest power at the sampling instant of bits one is obtained from the sampling of the bit one that comes after a long sequence of bits zero, marked with an empty circle in Figure 6b and c. On the other hand, the lowest power obtained from the sampling of the bit one comes after several consecutive ones, marked with a filled circle. This fact shows that the signal distortion due to the SOA nonlinearity is pattern-dependent, where each bit one is amplified with a gain that depends on the preceding bit sequences. As is shown in Figure 6b, when bit one arrives, providing its pattern preceding composed of a long sequence of zeros, the output power achieves its highest level. The output power reduces for the next arriving bits one, which is a decaying function of time starting from an initial value of power [43], which can be attributed to the carrier density depletion. The gain impact on signal quality in both linear and nonlinear regimes of SOA operation was investigated using Eqs. (17) and (18). By maintaining the device injection current constant, when the input signal power is low, SOA provides linear gain and high ASE. Whereas, gain and ASE reduce when the input power increases, thus affecting the signal Q-factor. Figure 7a shows the simulated gain as a function of input power for a wavelength of 1555 nm at variable injected currents such as 200, 92, and 70 mA corresponding at 30.9, 17.2, and 15.3 dB small-signal gain, respectively. Simulation results have been obtained using Eq. (3) under static operation conditions [41] for an SOA with 1800 $\mathrm{\mu }m$-length and 0.38 $\mathrm{\mu }m$-width active region. The arrows show saturation input power $P_{\text{in}}^{\mathrm{sat}}$, where the gain is reduced by 3-dB, for various bias currents. The analytical results of the $Q_b$and $Q_w$for the 200 mA driving current are presented as a function of gain in Figure 7b, reminding that the decreasing gain is due to the increasing input power. It is clear from the results that the signal quality increases while the gain decreases. The reason for this behavior is the ASE impact reduction on the signal stream at lower gain. Since ASE is a function of the gain, the signal performance is limited by ASE noise at high gain (low input power), while at a low gain (high input power), the signal performance is limited by patterning. In this case, the signal distortion at high input power has not severe effect on signal performance due to infinite ER assumptions. Moreover, the impact of bias current on the device gain and consequently on signal quality was investigated. The injected current was reduced from 200 mA down to 92 mA (17.2 dB gain) and 70 mA (15.4 dB gain).

Figure 6.

(a) Input signal example with infinite extinction ratio. Distorted output: (b) without ASE, (c) with ASE effect. Sampled values are shown with the dotted lines. The highest and the lowest output power at the sampling are marked by the dash-dotted and dashed lines, respectively. At the sampling instant of bits one, the highest power is marked by an empty circle, the lowest power is marked by a filled circle.

Figure 7.

(a) Simulated gain as a function of input power at I_dc applied in (b), (c), and (d). Arrows show the corresponding$P_{\text{in}}^{\mathrm{sat}}$at each I_dc. Quality factor as a function of SOA gain at different drive current: (b) I_dc=200 mA, G₀=30.9 dB and Q at$P_{\text{in}}^{\mathrm{sat}}$: Q_b=8.8 dB, Q_w=6.9 dB (c) I_dc=92 mA, G₀=17. 2 dB and Q at$P_{\text{in}}^{\mathrm{sat}}$: Q_b =19 dB, Q_w =18.3 dB (d) I_dc=70 mA, G₀=15.3 dB and Q at$P_{\text{in}}^{\mathrm{sat}}$: Q_b=20 dB, Q_w =19.4 dB. The Q_b and Q_w at the$G_{\mathrm{sat}}$marked by the dotted lines.$G_{\mathrm{sat}}$is the 3dB gain saturation at the saturation input power$P_{\text{in}}^{\mathrm{sat}}$.

Results are shown in Figure 7c and d, respectively. In the linear regime, as gain remains almost constant (∼G₀) the ASE contributions on $I_w^H$and $I_b^H$will be equal, which leads to identical results for both best and worst quality factor approximation $\left(Q_b=Q_w\right)$. Whereas, as the gain starts to saturate (nonlinear regime), the impact on signal quality gets intense at high injected current (200 mA), which translates into a small decrease of the $Q_w$value (Figure 7b). The signal quality at a saturation input power $P_{\text{in}}^{\mathrm{sat}}$improves at the lower injected current, where the gain value is smaller and carrier lifetime is longer. Therefore, whenever gain decreases, either by reducing injected current or increasing input power, the signal quality increases. It is worth mentioning that reducing injected current leads to decreasing G₀ and relatively decreases G_s, while increasing input power leads to decreasing G_s, but G₀ remains unchanged (without consideration ER effects). Accordingly, both small-signal gain and saturated gain have an impact on Q-factor. Indeed, the contribution of G₀ and G_sin Eqs. (17) and (18) are also through the P⁰_ASE and P^S_ASE, respectively. In another way, the ASE noise affects the optical signal-to-noise ratio (SNR), particularly at low input power where ASE is higher. Thus, at high gain, high ASE, the signal performance degrades (lower Q-factor). We noticed that, at low bias current and high input power, the obtained $Q_w$value is slightly higher than the $Q_b$. We attributed this increase of $Q_w$to the reduction of the ASE impact on $I_w^H$, since $I_w^H$corresponds to a lower gain, while $I_b^H$is given by the maximum value of the SOA gain, i.e., G₀.

It should be noticed, in this case, that for isolating the gain effect, we ignored all other effects on the signal quality that led to obtaining quality factors higher than realistic cases.

3.2 Extinction ratio

One of the most significant parameters that should be considered for a launched signal through an SOA is the signal ER, as it affects the amplified signal quality. When a binary pattern with finite ER propagates through an SOA, amplification occurs on both input bit levels. So that, not only bits one but also bits zero instantaneously experience amplification with the gain that depends on their former bit sequences (Figure 8). The highest power for bit zero is obtained when the pattern preceding bit zero is composed of several consecutive zeros, large enough for complete gain recovery, and can be approximated by $P_{\text{in}}^H{G}_0/\mathrm{ER}$. Furthermore, the lowest power is obtained when SOA reaches saturation, and the pattern-preceding bit zero is composed of several consecutive ones. The lowest power for bits zero can be approximated by $P_{\text{in}}^H{G}_s/\mathrm{ER}$. Following Eqs. (17) and (18), the best and worst quality factors in the case of limited extinction ratio will define as:

Figure 8.

Part of the simulated distorted output of an ideal rectangular input with an extinction ratio of 10 dB. Arrows show best and worst cases for low and high noise variances. Solid and dashed circles indicate the amplified zero level results of saturated gain and full recovery gain respectively.

$$Q_w=\frac{I_w^H-I_w^L}{{\sigma }_L^w+{\sigma }_H^w}=\frac{P_{\text{in}}^H{G}_s-\left[\left(P_{\text{in}}^H/\mathrm{ER}\right)G_0\right]}{\sqrt{{\left(P_{\mathrm{ASE}}^0\right)}^2+2P_{\mathrm{ASE}}^0\left(P_{\text{in}}^H/\mathrm{ER}\right)G_0}+\sqrt{{\left(P_{\mathrm{ASE}}^s\right)}^2+2P_{\mathrm{ASE}}^s{P}_{\text{in}}^H{G}_s}}$$E19

and,

$$Q_b=\frac{I_b^H-I_b^L}{{\sigma }_L^b+{\sigma }_H^b}=\frac{P_{\text{in}}^H{G}_0-\left[\left(P_{\text{in}}^H/\mathrm{ER}\right)G_s\right]}{P_{\mathrm{ASE}}^s+\sqrt{{\left(P_{\mathrm{ASE}}^0\right)}^2+2P_{\mathrm{ASE}}^0{P}_{\text{in}}^H{G}_0}}$$E20

In Figure 8, part of the simulated output signal pattern of an ideal rectangular input signal, with an extinction ratio of 10 dB, is represented. Four discrete values of best and worst case, for low and high noise variances, are marked by the arrows. Referring to Figure 8, it is evident, when the ER is finite, the low (‘zero’) level will be amplified too. Besides, its output power is variable due to gain variation from saturated gain (G_s) to full recovery gain (G₀). Given that, the eye-opening of the signal is affected by patterning and, as a consequence, the quality factor reduces. When SOA operates in the nonlinear regime, transmitting a long sequence of bits leads to gain compression due to carrier depletion, and gain achieves its lowest value. While, after transmitting a long sequence of bits zero, the SOA’s carrier density, and therefore its gain, recover to their maximum achievable values. Following this, when bit one arrives, the output achieves its maximum $I_b^H$, the carrier depletion starts, subsequently, the output starts to decay (Figure 8).

Under static operation conditions, maximum SOA gain can be found as a function of the average input power $\overline{P_{\text{in}}}=\left[P_{\text{in}}^H+\left(P_{\text{in}}^H/\mathrm{ER}\right)\right]/2$for low ER. For high ER values (ER_h), the maximum gain value assumes the small-signal value of G₀, but when the ER is decreased (i.e., ER < 10 dB as in practical systems) to lower values (ER_l), the maximum gain is indicated as G_l≤ G₀. When ER is smaller than 10 dB, the fully recovered gain depends on the input power and ER (ER < 10 dB). Exploiting Eq. (3), the SOA gain (G_l0) can be found as a function of the average input power for ER < 10 dB (Eq. (22)). Gain equations, hence, will take the form:

$$G_0=exp\left(\mathit{gL}\right)$$E21

$$G_{l0}=G_{0}exp\left\{\left(1-G_{l0}\right)\frac{\left[P_{\text{in}}^H+\left(P_{\text{in}}^H/\mathrm{ER}\right)\right]}{2P_{\mathrm{sat}}}\right\}$$E22

$$G_l=\frac{G_0+G_{l0}}{2}$$E23

$$G=G_{0}exp\left[\left(1-G\right)\frac{P_{\text{in}}}{P_{\mathrm{sat}}}\right]$$E24

where $g$is the net gain per unit length and $L$is the active region length. Being $P_{\mathrm{sat}}=\mathit{h\nu }{\sigma }_m/\Gamma {\tau }_{c}a$the saturation power of the SOA, where $\nu$is the optical frequency, $\Gamma$is the optical confinement factor, $a$is the differential gain coefficient, and ${\sigma }_m=W·d$is the active region cross-section, where $W$and $d$are its width and thickness, respectively. Eq. (22) can be solved numerically to obtain the full recovery gain G_lat ER_l(Eq. (23)). Thus, the average of G_l0and SOA small-signal gain (G₀) results in an estimated full recovery gain value for low ERs. This value is used as full recovery gain in Eqs. (19) and (20) when ER is low. Results are shown in Figure 9, where the maximum gain is calculated as a function of the input power for two ER values (ER_h= 20 dB (G₀curve (a)), and ER_l= 6 dB (G_lcurve (b))) and compared to the gain saturation curve (c). As it is illustrated in curve (b) of Figure 9, the fully recovered gain (G_l) for ER_l= 6 dB changes with input power. So that, at low input power, G_l(curve (b)) is close to the small-signal gain G₀(curve (a)), but by increasing input power, it gets spacing, where the distance depends on the input power and ER_l. When the ER is low (ER<10 dB), the impact of the amplified bits zero at any input power level, high or low, on signal quality is significant. For instance, at an input power of –12 dBm, the saturated gain is equal to 12.3 dB for both cases, high and low ERs, as is shown in Figure 9 (marked by a star). However, the value of the full recovery gain is related to the value of the input signal ER. Exploiting Eqs. (22) and (23) for ER = 6 dB, full recovery gain at the saturation input power (–12 dBm) obtained equal to 14.1 dB (marked by the square in Figure 9), which is 1.2 dB less than the full recovery gain at high ER (marked by a filled circle in Figure 9) achieved by Eq. (21). Employing these gains in Eqs. (19) and (20) lets us estimate the Q-factor. At confined ER, both bit levels (0s and 1s) are taking gain, and a number of the carriers will be used by $P_{\text{in}}^L$. The lower the ER(higher $P_{\text{in}}^L$), the more carriers will be taken by zero level power. This leads to reducing the number of carriers used by $P_{\text{in}}^H$and caused to change the value of full recovery gain, while at very high ERthe carriers will be fully available for $P_{\text{in}}^H$.

Figure 9.

Saturated amplifier gain G as a function of the input power for 15.3 dB of the small-signal gain. (a) Full recovery gain for input signals with high ER >10 dB. Obtained byEq. (21). The value of full recovery gain is constant and it is equal to small-signal gain for any input power and ER>10 dB, and (b) Full recovery gain for input signals with low ER ≤ 10. Obtained by Eqs. (22) and (23). The value of full recovery gain changes by changing ER (for ER ≤ 10) and it depends on the value of input power. The graph in (b) Obtained for ER = 6 dB. (c) Saturated amplifier gain, which obtains equally for both high and low ER usingEq. (24). (${\mathrm{I}}_{\mathrm{dc}}=70\mathrm{mA},\mathrm{\lambda }=1555\mathrm{nm}$). The fully recovered gain for ER = 6 dB and ER = 20 dB and the saturated gain, for both cases, at –12 dBm input power are marked by square, filled circle, and star, respectively.

Eventually, for low ER input signals, in Eqs. (19) and (20), G_lmust be used instead of G₀. The analytical results of Q-factor behavior versus input power are represented in Figure 10. These results are obtained with various ERs for an SOA with a linear small-signal gain of G₀ = 15.3 dB and saturation input power of –13 dBm. At high ER (20 dB) and low launch powers, the $Q_w$and $Q_b$have similar behavior (as explained above), while at high input powers, they act differently. By increasing input power, the gain decreases, and consequently, ASE noise decreases, the $Q_w$and $Q_b$increase accordingly (Figure 10a). By decreasing, ERinduces intensive patterning, thus the behavior of $Q_w$changes (Figure 10b) and it turns to a parabolic curve. In fact, in the linear regime, the ASE noise dominates, while in the nonlinear regime, patterning becomes the main signal degradation effect. Moreover, a further ER decrease introduces higher noise contribution in $Q_w$, which ends in reducing $Q_w$in linear operation area Figure 10c and d.

Figure 10.

Quality factor as a function of input power with, (a) ER = 20 dB, (b) ER = 15 dB, (c) ER = 8 dB, and (d) ER = 5 dB with small-signal gain G₀ = 15.3 dB.

At low ER and high input power, where the $Q_w$flattened, the SOA reaches the transparency point, where the absorptions (losses) and emissions (gain) are identical within the SOA. Input power beyond this point drives the SOA below the transparency, where it will not be able to recover its gain. In practice, at this point (i.e., material gain transparency [44]), the value of the quality factor will be unmeasurable and useless. Figure 11a and b show the contour plot for the calculated $Q_b$and $Q_w$versus the input power and a large range of ER. The plot was obtained using our analytical model for an SOA with the small-signal gain of G₀ = 15.3 dB at the wavelength of 1555 nm and injected current of 70 mA. The given boundary condition is equal to the Q-factor of 15.6 dB, corresponding to a BER of 10⁻⁹, and it is marked with the solid line. This enables the evaluation of the impact of the signal extinction ratio within different input powers on the amplifier performance. Eventually, the Q-factor is higher at the high extinction ratio signals, particularly on the high input power side. The SOA parameters used in our model have been derived from the characterization measurement of commercial SOA used in our experimental setup.

Figure 11.

Range of signal performance dependence on the input power and various ER for an SOA with an unsaturated gain ofG₀ = 15.3 dB. Lines separate the boundary with different quality factors in the (a) best (Q_b) and, (b) worst (Q_w) cases. The solid lines mark the boundary with error-free amplification of a data stream where theQ= 15.6 dB corresponds to a BER of 10⁻⁹.

3.3 Filtering

As it is shown in Figure 1b, the transmission system includes two bandpass filters, the first one, in the optical domain, usually placed just before photodetector, whose role is to remove out-band noise (such as ASE noise or cross talk due to adjacent channels). The second one in the electrical domain is generally matched to the transmission signal bandwidth to maximize the signal-to-noise ratio before the sampler. In a typical case of OOK NRZ transmission, a 4th order Bessel-Thomson filter with a bandwidth of 75% the bit rate is used. However, when nonlinearities affect the signal, such as SOA gain saturation, the matched filter does not provide optimal performance.

In this work, we performed analytical modeling of electrical filter effects on the Q-factor of the data streams and evaluated them through numerical simulation. At the first stage, we removed the SOA and optical filter. The effect of the electrical filter was considered on the received rectangular bit pattern. Then it was extended to investigate the electrical filter effects on the amplified bit pattern while SOA operated at the nonlinear regime. The numerical simulation of the filter effect on the data pattern and eye diagram of the rectangular signal is depicted in Figure 12. Applying the filter shapes the signal and affects the power of bits zero. Taking into account these facts, the effect of a Gaussian filter on signal quality was investigated. The performance of the Gaussian filter has also been compared with other conventional filters such as Super Gaussian and Raised Cosine (RC) filter. At the receiver input, an electrical Gaussian filter is included, where its 3-dB bandwidth is set to a frequency of 75% of the transmission data rate. The impulse response for the low-pass Gaussian filter, using Gaussian function, is defined as [45]:

Figure 12.

(a) Simulated filter effect on rectangular data pattern, (b) eye diagram of unfiltered signal, and (c) filtered by G₁. Arrows show normalized filter value at the end of the duration of the bit one (0.5) and in the sampling instant of the bit zero (0.062). ISI effect on bit zero is marked with a square.

$$h\left(t\right)=exp\left(-t^{2m}/2{\sigma }^{2m}\right)$$E25

where $\sigma$is the filter bandwidth, and it can define based on the bit time $\left(\sigma =\alpha T_b\right)$, and mis the filter order. Where m= 1 refers to the Gaussian filter, m> 1 refers to the Super Gaussian filter. Applying a first-order Gaussian filter (G₁) on the rectangular data stream, which would be applied to each symbol, makes each rectangular symbol becomes Gaussian-like. For the sake of simplicity, we normalized the signal power to unity. In this case, the normalized filter value (F) on the adjacent bit, when the bit is zero, can be calculated by taking the assumption of the filter value at the end of the duration of the preceding bit one be equal to 1/2, marked by a filled circle in Figure 12a. Considering the time interval from the second half of bit one to the first half of bit zero, the impulse response at T_b/2 would be $h\left(t-T_b/2\right)=1/2$, using Eq. (25) the $\alpha$, and subsequently, the filter bandwidth $\left(\sigma =0.42T_b\right)$is obtained. The value of this filter in the sampling instant of the adjacent bit zero is F_G1 = 0.062, marked by an empty circle in Figure 12a. We must notice that this filter value is valid when there is no Inter Symbol Interference (ISI) between two consecutive bits. The electrical filter can induce the ISI, where the bit gets smoothed out, and its energy spills over into the adjacent bits. The ISI causes a severe filter effect on bits zero and modifies the filter value on that, as it is shown in Figure 12a marked with a square. However, we demonstrate that using the filter value in the non-ISI case is enough to have a good estimation for the Q-factor. Figure 12a shows that one-level bits have different values. The reason for that can be found in the filter bandwidth. Implementing the same procedure, filter values at the adjacent bit sampling time for the second-order Gaussian (G₂) and Raised cosine (RC) filters are F_G2 = 0.183 and F_RC= 0.001, respectively. The impulse response of the low-pass RC filter [46] is given by:

$$h\left(t\right)=\frac{sin\left(t/T_b\right)}{t/T_b}\frac{cos\left(\mathit{\pi \alpha t}/T_b\right)}{1-{\left(2\mathit{\alpha t}/T_b\right)}^2}$$E26

where $\alpha =0.9$is the roll-off factor. Due to the analytically obtained low filter value at the sampling instant of bits zero for RC (F_RC = 0.001), the RC is estimated to provide an open eye diagram. The effect of the G₂ and RC filters on eye-opening are simulated and depicted in Figure 13. The filter values obtained through the analytical model are validated numerically. As a further step, the filter effect on amplified signal is investigated analytically, and the results are evaluated through numerical analysis. In order to isolate the filtering effect, we assume that the filter acts on the received bit stream represented by a rectangular bit pattern with a very high extinction ratio (37 dB) propagated through an SOA. In this case, best and worst quality factors can be expressed as:

Figure 13.

Simulated eye diagrams of filtered rectangular signal with (a) RC, (b) G_2.

$$Q_w=\frac{I_w^H-I_w^L}{{\sigma }_L^w+{\sigma }_H^w}$$

where,

$$I_w^H=P_{\text{in}}^H{G}_s,I_w^L=\left(\left(P_{\text{in}}^H/\mathrm{ER}\right)G_0+P_{\text{in}}^H{G}_{0}F\right),$$

$${\sigma }_L^w=\sqrt{{\left(P_{\mathrm{ASE}}^0\right)}^2+2P_{\mathrm{ASE}}^0\left(P_{\text{in}}^H/\mathrm{ER}\right)G_0+2P_{\mathrm{ASE}}^0{P}_{\text{in}}^H{G}_{0}F},$$

$${\sigma }_H^w=\sqrt{{\left(P_{\mathrm{ASE}}^s\right)}^2+2P_{\mathrm{ASE}}^s{P}_{\text{in}}^H{G}_s}$$E27

and,

$$Q_b=\frac{I_b^H-I_b^L}{{\sigma }_L^b+{\sigma }_H^b}$$

where,

$$I_b^H=P_{\text{in}}^H{G}_0,I_b^L=\left(\left(P_{\text{in}}^H/\mathrm{ER}\right)G_s+P_{\text{in}}^H{G}_{s}F\right),{\sigma }_L^b=P_{\mathrm{ASE}}^s,{\sigma }_H^b=\sqrt{{\left(P_{\mathrm{ASE}}^0\right)}^2+2P_{\mathrm{ASE}}^0{P}_{\text{in}}^H{G}_0}$$E28

As an example, we consider an SOA with a small-signal gain of 17.2 dB ($P_{\text{in}}^{\mathrm{sat}}$= –13.8 dBm) at the operating wavelength of 1555 nm. The analytical results while applying different EL filters are depicted in Figure 14. It is clear that the filter impact is more intense when the signal is affected by SOA nonlinearities. Therefore, in this operation condition, an optimal filter bandwidth may be very different from the standard matched one.

Figure 14.

Quality factor as a function of input power for 1555 nm signal, (a) without EL filter, (b) with RC filter, (c) with filter G₁, and (d) with filter G₂.

The model including different filters (RC, G₁, and G₂₎ have been validated numerically for an SOA input power large enough to assure a nonlinear regime (-4.8 dBm). We considered the effect of filters on signal quality by making a comparison of $Q_w$at a specific input power, –4.8 dBm, for each filter. The value of $Q_w$for the received signal without applying a filter is 26.4dB while it is reduced to 25.8 dB by applying the RC filter. It changes to 17.1, and 1 dB by applying G₁, G₂ filters, respectively. Such a huge change in $Q_w$highlight the importance of filter shape and parameters and demonstrate that matched filter used in the linear case may be not optimal. The analytically obtained results are in good agreement with the numerically simulated results (Figures 12 and 13).

As it is shown in Figure 14a, in the absence of an electrical filter, the signal quality at the high input power is high, even though the signal pattern in this area is affected by SOA nonlinearities. Figure 14b shows that the RC filter causes to reduce the $Q_w$at the high input powers, although RC induced lower effects than the other mentioned filters on signal quality. Referring to Figure 14c, applying G₁ on the received signal, leads to a decrease in the quality of the signal at the NL area of operation. This behavior is a consequence of the filter effects on the distorted signal. On the high input power side, the signal distortion occurs due to the patterning, however, the electrical filter induced the ISI between received symbols of the distorted signal. In the absence of the filter, the patterning does not cause to reduce the signal quality and there is no ISI between received symbols. Applying G₂ results in more severe ISI at high input powers and beyond a specific input power, the Q-factor becomes unmeasurable (Figure 14d). Since the value of the Super Gaussian filter on bits zero (F_G2) was found higher than the value of the Gaussian filter (F_G1), a higher ISI with G₂ was expected.

In conclusion, the electrical filter can be appropriate for optimizing performance in the linear operation to reduce the additive noise, however, for the nonlinear operation, it is inappropriate since it leads to a decrease in the Q-factor due to the patterning and ISI effects. Thus, system optimization must consider receiver filter shape and bandwidth as a function of SOA operation.

Finally, the performance of the G₁ on the amplified received signals, with various input power and different ERs, has been considered. In comparison with Figure 11, the effect of the filter on $Q_w$is more intense (Figure 15b while, as was expected from the analytical model, there is no significant quality degradation in$Q_b$. Hence, applying the G₁ electrical filter on the receiver side leads to degrading the signal quality on the high input power side due to the patterning and severe ISI (Figure 15b).

Figure 15.

First-order Gaussian filter effects on output signals of SOA with gain 15.3 dBm. Lines separate the boundary with different quality factors in the (a) best (Q_b), and (b) worst (Q_w) cases.

4. Experimental setup and measurement results

The analytical model is validated through experiments. For that purpose, SOA gain was characterized as a function of input power different bias currents (from 200 to 70 mA). Each bias current provides a specific G₀. Figure 16 shows the experimentally measured gain as a function of input power for a wavelength of 1555 nm for a driving current of 200 mA (92 mA). Simulation is included for comparison using the following SOA parameters: 1800 μm-length and 0.38 μm-width active region, G₀ = 30.9 dB (G₀ = 17.2 dB) at the 1555 nm wavelength, confinement factor of 0.4, and the transparency carrier density is$3.07\times {10}^{18}{\mathrm{cm}}^{-3}.$

Figure 16.

Gain as a function of input power at I_dc = 200 mA (I_dc = 92 mA). Simulation (solid line) and measured (dots). TheG₀and the${\mathrm{P}}_{\text{in}}^{\mathrm{sat}}$are 30.9 dB (17.2 dB) and –24 dBm (–14 dBm), respectively.

The experimental setup for model validation is depicted in Figure 17. A 10 Gb/s NRZ signal was generated by exploiting a Mach-Zehnder modulator (MZM) fed with a continuous wave laser (CW) at 1555 nm and driven by a bit pattern generator (BPG) with a 2¹¹-1 PRBS. The input power into the SOA is adjusted through a variable optical attenuator (VOA). The MZM bias has been adjusted to keep a constant extinction ratio of 13 dB for the generated rectangular signal. This tuning is performed to emulate the ER effects on the signal performance. The signal input power and extinction ratio had been measured before launching into the SOA. The amplifier is biased at 70 mA, which corresponds to 15.3 dB gain for 1555 nm wavelength with saturation input power of –13 dBm. A 0.7 nm optical band-pass filter (OBPF) was used to mitigate the emitted ASE noise for eye-diagram evaluation and Q-factor. The received amplified signal is detected utilizing a 40 GHz photo receiver followed by a 45 GHz real-time oscilloscope and a 10 Gb/s error detector. To investigate the ER effects, in both linear and NL operation regimes, the measurements have been done on the signal with different powers ranging between –20 dBm and +3 dBm, fed into the SOA. Figure 17 reports the eye diagram of the received signal in back-to-back and the eye diagram of the amplified signal, with +3 dBm injected power into the SOA. Although such a high input power drives the SOA into the deep saturation, the result shows a clear eye-opening with the quality factor of 16.4 dB (corresponding to a BER of 10⁻¹²), which is in good agreement with the analytically obtained result. The analytically obtained signal quality as a function of input power for a constant extinction ratio of 13 dB is depicted in Figure 18a together with the experimental measurement results. The analytically Q-factor for the signal with the input power of –11 dBm is 20 dB while in practice obtaining this value of Q-factor requires a device with very high sensitivity. Therefore, we considered the maximum sensitivity of the available device, Q= 16.9 dB, which corresponds to a BER of 10⁻¹² (dashed line in Figure 18) as the target for all obtained analytical quality factors which exceed 16.9 dB.

Figure 17.

Experimental setup for BER measurements of the amplified signal, while the input power is varied, the ER of the input signal to SOA is tuned to 13 dB (and 6 dB). The eye diagrams of the received signals are shown for the back-to-back (left) and after the SOA (right) with the input ER of 13 dB at 3 dBm input power. SOA drives at the bias current of 70 mA, which corresponds to a 15.3 dB gain for 1555 nm wavelength. BPG: bit pattern generator, OBPF: optical band-pass filter, PC: polarization controller, VOA: variable optical attenuator.

Figure 18.

Quality factor as a function of input power for 1555 nm signal with, (a) ER=13 dB, (b) ER=6 dB. The analyticalQ_bandQ_ware shown with the solid lines and dashed curves, respectively. Simulation and experimentally obtained results are marked. The dash-dotted lines show the Q of 12.6 dB (BER = 10⁻⁵), the dotted lines show the Q of 15.6 dB (BER = 10⁻⁹), the dashed lines show the Q of 16.9 dB (BER = 10⁻¹²).

To evaluate the model for low ER effects on SOA operation, we also performed the BER measurements for the signal transmission with ER = 6 dB and various powers at the SOA input. Experimental BER results are depicted in Figure 18b together with both analytical and simulated results. The analytical curves and simulation results are in good agreement with experimentally obtained results.

We made a comparison between a data pattern with a low and high extinction ratio. In Figure 19, the eye diagrams of amplified signals with –4.8 dBm input power, ER = 6 and 23 dB, are reported along with the corresponding data patterns. From the signal’s pattern, it is evident that low ER causes a severe patterning effect due to the distortion of zero level and a consequence eye degradation. While at high ER, there will be no ER effect on the zero level, and the patterning effect is only due to nonlinear effects.

Figure 19.

Eye diagrams of received amplified signal with SOA input power of –4.8 dBm and (a) ER = 23 dB, (b) ER = 6 dB, at a bias current of 70 mA, with the pattern of the corresponding signals (left side). The arrows marked the ER effect on zero level.

Based on the experimental results, we can interpret the behavior of the SOA quality factor due to our analytical model as follow. We estimated a boundary condition for the SOA quality factor $Q_b$, and $Q_w$. When SOA operates in the linear regime, the signal quality factor is compatible with the analytical $Q_b$, and when the input power increases, the signal quality tends to the $Q_w$_. At the input power above the SOA saturation input power (marked by an arrow in Figure 18), where SOA operates in the NL regime, the signal quality is consistent with the $Q_w$.

5. Discussion

The optimization of system performance requires proper knowledge of the device parameters, operating conditions, and system specifications when an SOA is used as a gain element. The availability of an analytical model capable of describing most of the dominant effects is beneficial in system design. Here, we proposed a method based on the analytical expressions for best and worst quality factors $\left(Q_b\text{and}{Q}_w\right)$. Within this model, the effect of variable intrinsic and extrinsic SOA factors, such as small-signal gain value, signal extinction ratio, and filtering, on signal quality factors has been investigated. The model was developed for NRZ modulation formats. It can afterward be improved for advanced modulation formats such as QPSK or M-QAM. Exploiting our model, we investigated the performance of an SOA with moderated gain through the analysis of the Q-factor by considering the influence signal extinction ratio. It has been noticed that using low gain SOA for low ERsignals gives a performance improvement. To evaluate the impact of the gain on the low ERsignals, we performed the contour plot for the Q-factor of an amplified signal over a wide range of input power at a low range of ER. SOA was driven with different currents (70 and 37 mA), which results in various small-signal gain at 1555 nm. The plot in Figure 20a shows the obtained $Q_w$at the small-signal gain of 15.3 dB and the results of 8.2 dB gain ($P_{\text{in}}^{\mathrm{sat}}$= –8.3 dBm) are depicted in Figure 20b. As represented in Figure 20, using SOA with the high gain for the low input ER causes degradation due to intense patterning.

Figure 20.

Signal performance dependence on the input power and ER for an SOA with a small-signal gain of (a) G₀ = 15.3 dB, and (b) G₀ = 8.2 dB. Lines separate the boundary with different quality factors (Q_w). The solid lines mark the boundary with error-free amplification of a data stream where theQ= 15.6 dB (BER of 10⁻⁹).

The following equations provide us with a guideline to estimate the Q-factor boundary conditions, with a different range of extinction ratios and a large range of SOA input power. This tool also leads us to choose a proper electrical filter at the receiver side:

$$Q_w=\frac{I_w^H-I_w^L}{{\sigma }_L^w+{\sigma }_H^w}$$

where

$$\begin{array}{c}{I}_w^H=P_{\text{in}}^H{G}_s\\ I_w^L=\left(\left(P_{\text{in}}^H/\mathrm{ER}\right)G_0+P_{\text{in}}^H{G}_{0}F\right)\\ {\sigma }_L^w=\sqrt{{\left(P_{\mathrm{ASE}}^0\right)}^2+2P_{\mathrm{ASE}}^0\left(P_{\text{in}}^H/\mathrm{ER}\right)G_0+2P_{\mathrm{ASE}}^0{P}_{\text{in}}^H{G}_{0}F}\\ {\sigma }_H^w=\sqrt{{\left(P_{\mathrm{ASE}}^s\right)}^2+2P_{\mathrm{ASE}}^s{P}_{\text{in}}^H{G}_s}\end{array}$$E29

and,

$$Q_b=\frac{I_b^H-I_b^L}{{\sigma }_L^b+{\sigma }_H^b}$$

where

$$\begin{array}{c}{I}_b^H=P_{\text{in}}^H{G}_0\\ I_b^L=\left(\left(P_{\text{in}}^H/\mathrm{ER}\right)G_s+P_{\text{in}}^H{G}_{s}F\right)\\ {\sigma }_L^b=P_{\mathrm{ASE}}^s\\ {\sigma }_H^b=\sqrt{{\left(P_{\mathrm{ASE}}^0\right)}^2+2P_{\mathrm{ASE}}^0{P}_{\text{in}}^H{G}_0}\end{array}$$E30

5.1 Lossless PICs

We propose to use this tool as guidelines to make a design for lossless photonic integrated circuits. There are plenty of research activities on the design and performance of PICs in the form of functional devices for standard networks and transmission. The standard networks require advanced PICs with the lowest loss. Several investigations have been done on providing new designs and technologies for the fabrication of low-loss PICs. However, using SOA in PICs, as a loss compensator can be an effective way of mitigating PICs losses. We propose the design guidelines for the SOA as an on-chip loss compensator, which provides a high-quality amplified signal to optimize the PIC performance. Exploiting the analytical model given in this manuscript, accompanied by numerical analysis of a multi-section cavity model we proposed in [41], can lead us to realize the optimum design of an SOA on-chip. For optimal SOA model design to incorporate the flip-chip bonding technique constraints in the hybrid integrated SOA–SOI chip, we proposed to exploit the multi-section analysis [41]. Our analytical model provides us with a guideline to easily design lossless PICs, using an SOA for loss compensation. Within this model, several effects, including reduced input extinction ratio, on SOA performance are considered. In addition, it was shown how these effects, can be compensated only by changing the SOA gain. Based on our analytical model, it is suitable to use a low gain SOA to ensure that signals are amplified with high quality, no matter the input signals extinction ratio is high or low. To illustrate that, we considered a PIC with an estimated total loss of about 10–12 dB. The estimated boundary conditions of quality factors [Eqs. (29) and (30)] guide us to choose an SOA with the proper gain for compensating the presented loss. Considering –9 dBm input signal, at the wavelength of 1555 nm with an extinction ratio of 5 dB, drives into the SOA.

As depicted in Figure 21, using 15 dB gain compensates 100% of loss at the cost of reducing Q-factor while using 10 dB gain, 70% of loss will be compensated but ensures an error-free operation. The reason behind the Q-factor reduction at this power, with 15 dB gain, is the OSNR degradation due to accumulated ASE noise emanating from the SOA. In our SOA model, the ASE noise is modeled as white Gaussian noise. Accordingly, we considered the affection of the accumulated noise from SOA on the Q-factor as noise variances. The effect of ASE in lower gain (10 dB) is smaller than that in higher gain (15 dB) at the input power of –9 dBm for 5 dB of ER. We should note that the –9 dBm input power is equal to the saturation input power of the SOA with 10 dB gain, and it drives the SOA with 15 dB gain into the nonlinear operation area. However, the saturated gain of the SOA, with 15 dB gain, at this input power is higher than the saturated gain of the other SOA (with 10 dB gain), which causes a higher ASE and a reduction in Q-factor. Another reason for the Q-factor reduction is the impact of the gain value on the zero-level bits. Since the ER is low, the higher gain has a severe impact on zero-level bits, consequent the Q-factor reduces. Although, at low input power both gains show the same behavior, though using low gain SOA is more appropriate at the higher input power. The results show only 10 dB SOA is enough for compensating 10 dB loss for both linear and nonlinear operation areas.

Figure 21.

Quality factor boundary of the SOA with G₀ = 15.3 dB (marked by stars) and saturation input power of –13 dBm, and the SOA with G₀ = 10 dB (marked by squares) and saturation input power of –9 dBm, for an input signal at 1555 nm with ER = 5dB.

5.2 Optimize SOA parameters

This tool also can be used as design guidelines to maximize the amplified signal Q-factor based on the SOA parameters. Designing an active region with a short length, large-cross section, and low confinement factor produces low G₀, high $P_{\text{in}}^{\mathrm{sat}}$, and large gain bandwidth [47]. Based on the type of the active region, the confinement factor value would be different, such that the quantum dot layers and, in most cases, the quantum wells have confinement factors lower than bulk materials. The amount of confinement factor influences the saturation power so that for a specified active region cross-section, decreasing confinement factor leads to an increase in the saturation power. It is appropriate to mention that $\mathrm{\Gamma }$is the fraction of the optical mode that overlaps with the active medium, therefore, a very low $\mathrm{\Gamma }$causes the signal to expand broadly out of the active region and leaks into the surrounding regions. After adapting the device geometry and material, choosing a proper bias current is required for obtaining the desired gain. For fixed geometrical parameters by defining the value of bias current, we can control the gain value, peak wavelength, injected carrier density, and carrier lifetime. Carrier lifetime is inversely proportional to the injected bias current, therefore, decreasing injected current increases carrier lifetime, which leads to a decrease in the saturation power. Furthermore, by decreasing bias current and, accordingly, carrier density decreases G₀and moves the peak to higher wavelengths.

It is worth mentioning, with a detailed perspective, we should consider that at very low currents, the stimulated emission (SE) dominates ${\tau }_c$, and at very large currents, the Auger recombination and spectral hole burning (SHB) lead to the different behavior of ${\tau }_c$. Where, for the reason of simplification, we approximated the carrier lifetime inversely proportional to the injected bias current. This simplification is concerning using only one rate equation to describe all carriers in the device [Eq. (31)] [40].

The boundary conditions of the quality factor show that the Q-factor behavior is related to the value of G₀and n_sp. Note that the P_ASE is proportional to the n_sp. Therefore, in addition to choosing the proper gain, optimizing the population-inversion factor leads to optimizing Q-factor. As an example of using this study as a guideline for designing SOA, we considered the performance of an SOA with different active lengths but with the same peak gain and peak wavelength and the same saturation power. Achieving this, we start considering a device with a small active length and the peak gain of about 8 dB at 1555 nm, then the length increases, and the bias current changes accordingly to attain the same peak gain as the device with the smaller length. All other device parameters, including the confinement factor, remain constant. In such a manner, the carrier density, therefore, the population-inversion factor, changes by changing the bias current and active length.

Initially, we modeled SOA based on the experimentally obtained and fit parameters, using the numerical model described in [41]. Afterward, we changed the active region length and some other parameters to achieve the ≈ 8 dB gain. We modeled SOAs with different active region lengths (L) while trying to keep G₀ and P_sat constant. Somehow, at large L, the injected carriers (n) by the same bias current are not enough to provide the same gain as the small length SOA. To overcome this fact, we increase I_dc with the proper value, which leads to injecting enough carriers to attain the same gain. The injected carriers’ value at unity quantum efficiency is given by [42].

$$n=\frac{I_{\mathit{dc}}{\tau }_c}{q{\sigma }_{m}L}$$E31

where $q$is the electron charge. The I_dc and Lincreased somehow to have lower nat larger Lthat corresponds to higher n_spat larger L, where $n_{\mathit{sp}}=\frac{n}{n-n_{\mathit{tr}}}$and $n_{\mathit{tr}}$is transparency carrier density. Observing Q-factor for different active region lengths, while the peak wavelength, G₀, and P_sat kept being constant, shows that the effects of SOA lengths on n_spimpact Q-factor (Figure 22). The parameters used in the simulations are presented in Table 1.

Figure 22.

Quality factor boundary of the linear operation area of an SOA with different active regime length, where its peak gain and the saturation power remains constant. The input signal power and extinction ratio are –15 dBm and 6 dB, respectively.

Symbol	Description	value	Unit
w	Width of active region	0.4	$\mathrm{\mu m}$
d	Thickness of active region	0.1	$\mathrm{\mu m}$
$\mathrm{\Gamma }$	Confinement factor	0.4	—
$n_{\mathit{tr}}$	Transparency carrier density	$3.08\times {10}^{18}$	${\mathrm{cm}}^{-3}$
$\alpha$	Internal loss coefficient	$3.5$	${\mathrm{cm}}^{-1}$
$a$	Differential gin coefficient	$1.7\times {10}^{-16}$	${\mathrm{cm}}^2$
$\overline{n}$	Average refractive index	$3.45$	-
${\tau }_c$	Free carrier life time	382	Ps
${\lambda }_p$	Peak wavelength	1555	nm

Table 1.

SOA geometrical and material parameters were used in the simulation.

With regards to Eqs. (29) and (30), while the extinction ratio of the input signal is high and there is no electrical filter on the receiver side (F= 0), the Q-factor will be influenced by the SOA parameters and the input power. In this example, we considered Q-factor behavior performance in the linear operation regime at the input power of –15 dBm with the extinction ratio of 6 dB. The length effect of the active region on the Q-factor behavior is depicted in Figure 22.

When the length decreased, the peak gain was maintained constant by reducing the bias current simultaneously. Decreasing bias current decreases the population-inversion factor, which results in optimized Q-factor boundary conditions. Thereby a device with low G₀ and low n_sp with short active region length is expected to work with error-free amplification. It should be mention that decreasing the length increases the gain bandwidth, therefore, while the peak wavelength could remain constant by reducing the bias current, the side wavelengths, far from the peak, might take different gain in different active region lengths.

6. Conclusion

We proposed an analytical model for estimating the performance of optical transmission systems including SOA in terms of Q-factor. The model allows to design high-performance SOAs based on system design constraints and requirements. In PIC with integrated SOAs, the model accompanied by the multi-section cavity model we proposed in [41] provides a tool for improving the design, limiting the number of fabrication runs and related costs. The model estimates the Q-factor based on SOA parameters (gain), signal properties (ER), and filter type at the receiver side. The effects of ASE noise and saturated gain compression are also included in the model. The model was validated through numerical simulation and experiments. The dependency of the optimal SOA gain on the signal extinction ratio has been investigated. This fact is a crucial aspect in optical interconnection applications that use low extinction ratio transmitters. It was shown at a low extinction ratio (ER< 10 dB), the fully recovered gain changes with input power and ER. Based on our model, we proposed the design of a lossless PIC, employing low gain SOAs for obtaining a high-quality amplified signal. It has been demonstrated that using low gain SOAs in PICs allow counteracting signal distortions, in both linear and nonlinear operation regime. In addition, we have studied the impact of different types of receiver filters on the performance of an amplified signal. It was shown that using a Gaussian filter leads to reduced signal quality due to patterning effects and ISI. Nevertheless, the impact of an RC filter is less intense, so confirming that the selection of a proper filter is a significant task while dealing with an amplified signal.

Acknowledgments

The authors thank Dr. Francesco Fresi for fruitful discussions and invaluable suggestions.

Conflict of interest

The authors declare no conflict of interest.