**All-optical Semiconductor Optical Amplifiers Using Quantum Dots (Optical Pumping)**

Khalil Safari, Ali Rostami, Ghasem Rostami and Mahboubed Dolatyari

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62107

#### **Abstract**

[76] H. Wessing, et al., "Combining Control Electronics with SOA to Equalize Packet-to-Packet Power Variations for Optical 3R Regeneration in Optical Networks at 10

[77] A.V. Tran, et al., "Optical Packet Power Equalization with Large Dynamic Range us‐

[78] N. Cheng, et al., "Long Reach Passive Optical Networks with Adaptive Power Equal‐ ization using Semiconductor Optical amplifiers," OSA/ACP'2009, paper FS4, 2009. [79] Y. Kai, et al., "MSA Compatible Size, Dual-channel Fast Automatic Level Controlled SOA substems fro Optical Packet and PON signals," in Proc. ECOC'2011, Mo.2.A.2,

[80] G. de Valicourt, et al., "Distributed fast optical packet power equalization for effi‐ cient WDM packet switched networks," in Proc. ECOC'12, Tu.3.A.4, Amsterdam,

ing Controlled Gain-Clamped SOA," OFC 2005, OME46, 2005.

Gbit/s," OFC'2004, paper WD2, 2004.

84 Some Advanced Functionalities of Optical Amplifiers

2011.

Netherlands, 2012.

In the first portion of this chapter, a short review on all-optical processing is pre‐ sented. Following the ideas of all-optical processing, a basic unit cell is introduced for the realization of these systems. To this end, an all-optical semiconductor optical amplifier based on quantum dots (QD-SOA) is presented and used as the basic unit cell. Then, a novel scheme for a high-speed all-optical half-adder based on quantum dot semiconductor optical amplifiers has been theoretically and extensively ana‐ lyzed. We accelerate the gain recovery process in QD-SOA with a control pulse (CP) using the cross-gain modulation (XGM) effect in QD-SOA (based on a novel work reported by Rostami *et al* published in *IEEE J. Quantum Electron* in 2010). In this pro‐ posed scheme, a pair of input data streams simultaneously drives the switch to pro‐ duce sum and carry. The proposed scheme is driven by the pair of input data streasms for one switch between which the Boolean XOR function is to be executed to produce a sum-bit. Then, one of the input data is utilized to drive the second switch and another is used as input data for it to produce a carry-bit. In the pro‐ posed structure, we need to use an optical attenuator to reduce the power level of the optical signal. Thee, data pulse is at least an order of magnitude stronger than the incoming pulse; thereforehowever, only the input pulse can alter QD-SOA's op‐ tical properties. Also, an all-optical cross-phase modulation (XPM) wavelength con‐ verter has been utilized to obtain an all-optical AND gate, which is logic CARRY. Logic SUM and CARRY are simultaneously realized in the proposed structure. The operation of the system is evaluated and demonstrated with a Tb/s bit rate. The pro‐ posed structure is mathematically modeled by writing rate equations and then is numerically simulated with success. High-speed operation capabilities of the pro‐ posed all-optical half-adder structure are evaluated by numerical simulation.

**Keywords:** All-optical Half-Adder, QD-SOA, Optical Pumping, High-speed Processing

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **1. Introduction**

In the past few years, all-optical signal processing concepts and technologies have evolved remarkably mainly due to the discovery of semiconductor-based all-optical switches. Elec‐ tronic devices such as switches and routers are not fast enough, whereas the speed of optical communication systems can reach up to 10 Tb/s [1]. Therefore, in order to realize these capabilities in optical engineering, recently, there washas been a huge motivation for research‐ ers in this area to concentrate on the implementation of the optical digital logic gates (e.g., AND, OR, NOT, and XOR) and optical logic modules (e.g., counters, adders, subtractors, and shift registers) [2]. In SOAs, nonlinear effects such as cross-gain modulation (XGM), crossphase modulation (XPM), four-wave mixing (FWM), and transient cross-phase modulation can all be exploited to demonstrate all-optical signal processing functions. Several schemes on all-optical processing/operations, including nonlinear materials, terahertz optical asymmetric demultiplexer (TOAD), cross gain modulation in a semiconductor optical amplifier, SOA– Mach–Zehnder interferometer (SOA–MZI), Pperiodically poled lithium niobate (PPLN) waveguide, and ultrafast nonlinear interferometer (UNI), were proposed [3]. Most of the proposed designs are based on two or more interferometric switches; a synchronization problem between different switches restricts their designs for practical implementation. QD-SOA has a remarkable ultrafast response, larger unsaturated gain, and a much faster gain recovery after gain compression than bulk SOAs, small electron relaxation times, highsaturation output power, low-noise figure, and large-gain-bandwidth product at the same time. One of basic demands for high-speed operation is high-speed carrier dynamics. In QD-SOA, there is a high-speed carrier transport, therefore high-bit-rate devices such as logic gates can be realized based on this device. It is clear that due to band offset between QDS and wetting layer (WL), capture time will be much shorter than this parameter in bulk and quantum well structures. This parameter is between femtoseconds (fs) and picoseconds (ps) in QDs and therefore this is the main reason for high-speed operation of QD-SOA-based devices and systems. The response time for gain saturation is 100 fs−1ps, which is sufficient for a gigabit to sub-terabit optical transmission system [4]. Technology of the quantum-dot SOAs is suitable owing to its remarkably ultrafast response, which, combined with its attractive characteristics, distinguishes them from conventional SOAs. Compared to other types of amplifiers, One of other properties of the QD-SOA is a large unsaturated gain versus other type of amplifiers, which thatleadss to higher power output. Therefore, in low-injection electric pump, there is sufficient optical gain and thus less power consumption. Also, in electric pump case, optical gain can be controlled by current density and in optical pumpcase using intensity of pump signal too. QD-SOA illustrates faster gain recovery time versus other types of optical amplifi‐ ers. Thus, QD-SOA can be used for high-speed optical signal processing without distortion [5]. The all-optical logic gates, optical wavelength conversion, and in this chapter, we develop a theoretical approach for compensation of the carrier relaxation time into excited state (ES). In our model, we have considered two energy levels in both conduction and valence bands. It will be shown that applying a CP with enough energy will highly accelerate the recovery process of QD-SOA and will lead to a high-bit-rate operation of QD-SOA–MZI structure in the presence of the CP. In recent years, the optical logic based on several different schemes has been demonstrated and reported, which is based on the dual-semiconductor optical amplifier Mach–Zehnder interferometer [6,7,8], semiconductor laser amplifier (SLA) loop mirror [5–8], ultrafast nonlinear interferometers [9–12], and four-wave mixing process in SOA [13] and other alternatives. In the past reported research results, there are several publications for the realization of all-optical half adders such as terahertz optical asymmetric demultiplexers and ultrafast nonlinear interferometers [14, 15]. Based on these excellent properties of QD-SOA, interferometric effect such as the Sagnac phenomenon was realized using this optical device [16–20]. Also, using nonlinear effects such as cross-phase modulation and cross-gain modu‐ lation, all-optical XOR and AND gates were implemented by Ki *et al* and finally the half-adder was implemented [16]. Considering the nonlinear properties of SOA, an all-optical half-adder operating in 10 Gbit/s was reported in Ref. [17]. Another proposal based on four-wave mixing in SOA was illustrated in Ref. [18]. The Hhalf-adder/subtractor unit based on dark–bright solitons has been reported in Ref. [19]. All-optical half-adder using planar three-core nonlinear directional coupler was presented in Ref. [20]. For completing all-optical half-adder, mono‐ lithically integrated SOA-based -MZI switches were used to provide compact size, thermal stability, high-speed compatibility, low switching energy, relative stability, and optical integration compatibility [21–23]. Here, we present a theoretical model of an ultrafast alloptical half-adder based on the two QD-SOA-based MZI, where, in the first switch, the pair of input data streams execute the Boolean XOR function to produce a sum-bit. Besides, an alloptical XPM wavelength converter has been utilized to obtain an all-optical AND gate, which is logic CARRY. Logic SUM and CARRY are simultaneously realized [24–26]. Our proposed design requires less number of switches and other auxiliaries as compared with the other halfadder circuits using semiconductor optical amplifier-based devices. The all-optical half-adder has the potential to execute the addition in the optical domain up to 2 Tb/s. The configuration of the proposed all-optical half-adders using two symmetrical QD-SOA-based MZI switches is presented as follows: Logic SUM and CARRY are simultaneously implemented to realize the all-optical half-adder. The all-optical half-adder uses only two input signals. In electrical engineering, all arithmetic-related functions including addition, subtraction, multiplication, and division are realized by two half-adders and are a prerequisite for all-optical processors [27]. Boolean functions of logic SUM and logic CARRY exactly coincide with the XOR gate and the AND gate.

**1. Introduction**

86 Some Advanced Functionalities of Optical Amplifiers

In the past few years, all-optical signal processing concepts and technologies have evolved remarkably mainly due to the discovery of semiconductor-based all-optical switches. Elec‐ tronic devices such as switches and routers are not fast enough, whereas the speed of optical communication systems can reach up to 10 Tb/s [1]. Therefore, in order to realize these capabilities in optical engineering, recently, there washas been a huge motivation for research‐ ers in this area to concentrate on the implementation of the optical digital logic gates (e.g., AND, OR, NOT, and XOR) and optical logic modules (e.g., counters, adders, subtractors, and shift registers) [2]. In SOAs, nonlinear effects such as cross-gain modulation (XGM), crossphase modulation (XPM), four-wave mixing (FWM), and transient cross-phase modulation can all be exploited to demonstrate all-optical signal processing functions. Several schemes on all-optical processing/operations, including nonlinear materials, terahertz optical asymmetric demultiplexer (TOAD), cross gain modulation in a semiconductor optical amplifier, SOA– Mach–Zehnder interferometer (SOA–MZI), Pperiodically poled lithium niobate (PPLN) waveguide, and ultrafast nonlinear interferometer (UNI), were proposed [3]. Most of the proposed designs are based on two or more interferometric switches; a synchronization problem between different switches restricts their designs for practical implementation. QD-SOA has a remarkable ultrafast response, larger unsaturated gain, and a much faster gain recovery after gain compression than bulk SOAs, small electron relaxation times, highsaturation output power, low-noise figure, and large-gain-bandwidth product at the same time. One of basic demands for high-speed operation is high-speed carrier dynamics. In QD-SOA, there is a high-speed carrier transport, therefore high-bit-rate devices such as logic gates can be realized based on this device. It is clear that due to band offset between QDS and wetting layer (WL), capture time will be much shorter than this parameter in bulk and quantum well structures. This parameter is between femtoseconds (fs) and picoseconds (ps) in QDs and therefore this is the main reason for high-speed operation of QD-SOA-based devices and systems. The response time for gain saturation is 100 fs−1ps, which is sufficient for a gigabit to sub-terabit optical transmission system [4]. Technology of the quantum-dot SOAs is suitable owing to its remarkably ultrafast response, which, combined with its attractive characteristics, distinguishes them from conventional SOAs. Compared to other types of amplifiers, One of other properties of the QD-SOA is a large unsaturated gain versus other type of amplifiers, which thatleadss to higher power output. Therefore, in low-injection electric pump, there is sufficient optical gain and thus less power consumption. Also, in electric pump case, optical gain can be controlled by current density and in optical pumpcase using intensity of pump signal too. QD-SOA illustrates faster gain recovery time versus other types of optical amplifi‐ ers. Thus, QD-SOA can be used for high-speed optical signal processing without distortion [5]. The all-optical logic gates, optical wavelength conversion, and in this chapter, we develop a theoretical approach for compensation of the carrier relaxation time into excited state (ES). In our model, we have considered two energy levels in both conduction and valence bands. It will be shown that applying a CP with enough energy will highly accelerate the recovery process of QD-SOA and will lead to a high-bit-rate operation of QD-SOA–MZI structure in the presence of the CP. In recent years, the optical logic based on several different schemes has

## **2. Principles and design of proposed all-optical half-adders**

The proposed structures for half-adders (schematics and SOA-based structures) are shown in Figs. 1 and 2. In order to implement the proposed half-adder in an optical domain, the alloptical XOR and AND gates are reviewed at first. To implement the all-optical XOR gate, two SOAs with cross-gain modulation are used. To implement the all-optical AND gate, binary characteristics of the XPM wavelength converter are used too. Since the operations of XOR and AND logic gates depend on XGM and XPM phenomenon in semiconductors, the maximum speed of half-adder can be increased up to the limit of XGM and XPM. The speed of operation for all-optical half-adders is limited by pulse width and recovery time in SOAs.

Fig. 1‐a. Basic structure of half‐adder [14]. Fig. 1‐b. Half‐adder without control pulse. **Figure 1.** (a). Basic structure of half-adder [14]. (b). Half-adder without control pulse.

(b)

Fig. 2‐a. Configuration of the proposed all‐optical half‐adder using two symmetrical QD‐SOA‐based MZI switches. An electronic circuit (combinational circuits) performing addition of two binary digits is denoted as a half-adder. The carry bit will be 1 if both bits are 1, else will be zero. The sum-bit is the most significant bit of addition. A schematic of all-optical half-adders is illustrated in Fig. 2. It includes two MZIs in which QD-SOAs are in arms. For the first MZI, data inputs are A and B. Depending on data values A and B, the incoming pulse reaches Port-1 or Port-2. In this structure, output Port-1 corresponds to XOR operation and Port-2 corresponds to XNOR operation [3]. One of the attenuated input data acts as an incoming pulse for the second switch (MZI-2) and the other input data A (or B) produces the output of carry-bit. In other words, the output Port-1 of the second switch gives the logic operation AND. The four possible cases are described as follows: the input data are 1 or zero when the light beam is present or absent, respectively. Based on Fig. 2, when A=B=0, the IP signal is applied only on the first MZI. Thus, we have the signal only from Port-2 and, Port-1 will be zero and Port-2 will be 1 [3]. Therefore, Port-1 becomes inactive. It generates the output as 0, that is, carry=0. In the first switch (MZI-1), if one of the input signals (either A or B) is 0 while the other is 1, the incoming pulse emerges from the Port-1. Thuserefore, the sum (S) will be equal to 1. Now, in the second switch (MZI-2),

Fig. 1‐a. Basic structure of half‐adder [14]. Fig. 1‐b. Half‐adder without control pulse.

(a)

88 Some Advanced Functionalities of Optical Amplifiers

(b) Fig. 1‐a. Basic structure of half‐adder [14]. Fig. 1‐b. Half‐adder without control pulse.

An electronic circuit (combinational circuits) performing addition of two binary digits is denoted as a half-adder. The carry bit will be 1 if both bits are 1, else will be zero. The sum-bit is the most significant bit of addition. A schematic of all-optical half-adders is illustrated in Fig. 2. It includes two MZIs in which QD-SOAs are in arms. For the first MZI, data inputs are A and B. Depending on data values A and B, the incoming pulse reaches Port-1 or Port-2. In this structure, output Port-1 corresponds to XOR operation and Port-2 corresponds to XNOR operation [3]. One of the attenuated input data acts as an incoming pulse for the second switch (MZI-2) and the other input data A (or B) produces the output of carry-bit. In other words, the output Port-1 of the second switch gives the logic operation AND. The four possible cases are described as follows: the input data are 1 or zero when the light beam is present or absent, respectively. Based on Fig. 2, when A=B=0, the IP signal is applied only on the first MZI. Thus, we have the signal only from Port-2 and, Port-1 will be zero and Port-2 will be 1 [3]. Therefore, Port-1 becomes inactive. It generates the output as 0, that is, carry=0. In the first switch (MZI-1), if one of the input signals (either A or B) is 0 while the other is 1, the incoming pulse emerges from the Port-1. Thuserefore, the sum (S) will be equal to 1. Now, in the second switch (MZI-2),

**Figure 1.** (a). Basic structure of half-adder [14]. (b). Half-adder without control pulse.

Fig. 2‐a. Configuration of the proposed all‐optical half‐adder using two symmetrical QD‐SOA‐based MZI switches.

gate [25]. **Figure 2.** (a). Configuration of the proposed all-optical half-adder using two symmetrical QD-SOA-based MZI switches. (b). XOR gate [25].

AND gate produces the output as 0, that is, carry=0. When the input signal is 1 (A=B=1), the IP signal is applied on the first MZI as well. In this case, Port-2 will be 1. This way, the circuit performs the addition operation between two-bit binary data. In the next section, these theoretical results are verified through numerical simulations using Matlab [3]. An electronic circuit (combinational circuits) performing addition of two binary digits is denoted as a half‐adder. The carry bit will be 1 if both bits are 1, else will be zero. The sum‐bit is the most significant bit of addition. A schematic of all‐optical half‐adders is illustrated in Fig. 2. It includes two MZIs in which QD‐SOAs are in arms. For the first MZI, data inputs are A and B. Depending on data values A and B, the incoming pulse reaches Port‐1 or

Port‐2. In this structure, output Port‐1 corresponds to XOR operation and Port‐2 corresponds to XNOR operation

#### **3. Operational principles of QD-SOA-based MZI switch** [3]. One of the attenuated input data acts as an incoming pulse for the second switch (MZI‐2) and the other input

The ease of manufacturing, installation, and operation of all-optical signal processing and communication needs integrated optics-based devices and systems [28]. As an example, the MZI is one of integrated optical building block and one can make SOA on. Thus, MZI-based SOA will be one of the integrated optical basic blocks [28]. Considering the basic principles of SOA–MZI for operation, logic gates show that it is well known when XPM and XGM or other nonlinear phenomenon in SOA is used while it is inserted in arms of MZI. In Fig. 3, a simple schematic of logic gates using SOA–MZI is shown, where SOA operates as a nonlinear element. In this structure, the pump light controls dynamics of nonlinear effects in SOA and then probe data A (or B) produces the output of carry‐bit. In other words, the output Port‐1 of the second switch gives the logic operation AND. The four possible cases are described as follows: the input data are 1 or zero when the light beam is present or absent, respectively. Based on Fig. 2, when A=B=0, the IP signal is applied only on the first MZI. Thus, we have the signal only from Port‐2 and, Port‐1 will be zero and Port‐2 will be 1 [3]. Therefore, Port‐1 becomes inactive. It generates the output as 0, that is, carry=0. In the first switch (MZI‐1), if one of the input signals (either A or B) is 0 while the other is 1, the incoming pulse emerges from the Port‐1. Thuserefore, the sum

(S) will be equal to 1. Now, in the second switch (MZI‐2), AND gate produces the output as 0, that is, carry=0. When

light propagates inside the nonlinear media and experiences nonlinear optical effects. On analyzing this structure, it is observed that the control and probe fields can return to zero or non-return to zero light. Based on carrier, depletion that is occurred in SOA will conclude to gain and phase modulation which named as XGM and XPM. It should be mentioned that these effects are so efficient and thus in SOA and QD-SOA it can be realized in very short length and low pump power. As we mentioned earlier, since optical pump light is amplified in SOA, therefore low pump power is required. As shown in Fig. 3, low-gain recovery time in SOAs can be compensated in the MZI- arrangement when SOAs are in both arms as differential form. In Fig. 4, total carrier variation in SOAs using symmetric- Mach-–Zehnder (SMZ) gate-based gaiting window is illustrated. Using a control pulse with given pulse duration, a repetition rate is introduced to make a periodic variation in carrier density. When a short pulse width is used, the carrier density is depleted and also slow recovery time can be compensated by exciting both arms in MZI with suitable delay time. Based on the proposed ideas, the rise and fall times are defined by the control pulse duration [28].

**Figure 3.** SMZ configuration and nonlinear phase response cancel out mechanism [26].

**Figure 4.** Total carrier density of the SOAs in the presence of control pulses (left) and SMZ gate output (right) [26].

#### **3.1. SOA–MZI transfer function**

light propagates inside the nonlinear media and experiences nonlinear optical effects. On analyzing this structure, it is observed that the control and probe fields can return to zero or non-return to zero light. Based on carrier, depletion that is occurred in SOA will conclude to gain and phase modulation which named as XGM and XPM. It should be mentioned that these effects are so efficient and thus in SOA and QD-SOA it can be realized in very short length and low pump power. As we mentioned earlier, since optical pump light is amplified in SOA, therefore low pump power is required. As shown in Fig. 3, low-gain recovery time in SOAs can be compensated in the MZI- arrangement when SOAs are in both arms as differential form. In Fig. 4, total carrier variation in SOAs using symmetric- Mach-–Zehnder (SMZ) gate-based gaiting window is illustrated. Using a control pulse with given pulse duration, a repetition rate is introduced to make a periodic variation in carrier density. When a short pulse width is used, the carrier density is depleted and also slow recovery time can be compensated by exciting both arms in MZI with suitable delay time. Based on the proposed ideas, the rise and

fall times are defined by the control pulse duration [28].

90 Some Advanced Functionalities of Optical Amplifiers

**Figure 3.** SMZ configuration and nonlinear phase response cancel out mechanism [26].

**Figure 4.** Total carrier density of the SOAs in the presence of control pulses (left) and SMZ gate output (right) [26].

MZIs with SOAs on arms are the most suitable structures for applications such as optical logic gates. Considering similar applications such as other fiber-based devices, including SOAs, and nonlinear elements, such as semiconductor laser amplifier loop optical mirror (SLALOM) and terahertz optical asymmetric demultiplexer, the SOA in MZI structure presented in Fig. 5 can be modeled with a nonlinear device with a gain effect and a phase shift applied on the input signal [28]. Therefore, the transfer function of the considered structure can be derived in the the following manner. In the above configuration, (*A*1, *A*2) and (*D*1, *D*2) are input and output lightwaves, respectively, (*k*1, *k*2) are normalized coupling coefficients of the input and output couplers and (*B*1, *B*2) and (*C*1, *C*2) are input and output lightwaves of the SOAs, respectively. The gain and phase shift of each of the SOAs are considered with (G1, *φ*1) and (G2, *φ*2) for upper and lower arms of SOAs. Thus, considering both pairs of input and output lightwaves, the following coupled equations can be obtained [28]:

$$
\begin{pmatrix} B\_1 \\ B\_2 \end{pmatrix} = \begin{pmatrix} \cos K\_1 & i \sin K\_1 \\ i \sin K\_1 & \cos K\_1 \end{pmatrix} \begin{pmatrix} A\_1 \\ A\_2 \end{pmatrix} \tag{1}
$$

We assumed that an optical signal travelling through the amplifier would experience an amplification of *G* gain and a phase shift of *φ*. Therefore,

$$
\begin{pmatrix} \mathbf{C}\_1 \\ \mathbf{C}\_2 \end{pmatrix} = \begin{pmatrix} \sqrt{\mathbf{G}\_1} e^{i\Phi\_1} & 0 \\ 0 & \sqrt{\mathbf{G}\_2} e^{i\Phi\_2} \end{pmatrix} \begin{pmatrix} B\_1 \\ B\_2 \end{pmatrix} \tag{2}
$$

Then, the transfer function can be described as

$$
\begin{pmatrix} D\_1 \\ D\_2 \end{pmatrix} = \begin{pmatrix} H\_{11} & H\_{12} \\ H\_{21} & H\_{22} \end{pmatrix} \begin{pmatrix} A\_1 \\ A\_2 \end{pmatrix} \tag{3}
$$

where

$$\begin{aligned} H\_{11} &= \cos K\_1 \cos K\_4 \sqrt{G\_1} e^{i\phi\_1} - \sin K\_1 \sin K\_4 \sqrt{G\_2} e^{i\phi\_2} \\ H\_{21} &= i(\cos K\_1 \sin K\_4 \sqrt{G\_1} e^{i\phi\_1} + \sin K\_1 \cos K\_4 \sqrt{G\_2} e^{i\phi\_2}) \\ H\_{21} &= i(\cos K\_1 \sin K\_4 \sqrt{G\_1} e^{i\phi\_1} + \sin K\_1 \cos K\_4 \sqrt{G\_2} e^{i\phi\_2}) \\ H\_{22} &= -\sin K\_1 \sin K\_4 \sqrt{G\_1} e^{i\phi\_1} + \cos K\_1 \cos K\_4 \sqrt{G\_2} e^{i\phi\_2} \end{aligned} \tag{4}$$

**Figure 5.** Schematic of SOA-incorporated MZI structure [26].

Denoting the input and output signal powers with PA1, PA2, PD1, and PD2 and assuming an ideal 3 dB coupler (sin*ki* = 2 / 2, cos*ki* = 2 / 2), the transfer function reduces to

$$H\_{D1} = \frac{P\_{D1}}{P\_{A1}} = \frac{1}{4}G\_1 + \frac{1}{4}G\_2 - \frac{1}{2}\sqrt{G\_1 G\_2} \cos \Delta \Phi \tag{5}$$

$$\begin{aligned} H\_{D2} &= \frac{P\_{D2}}{P\_{A1}} = \frac{1}{4}G\_1 + \frac{1}{4}G\_2 + \frac{1}{2}\sqrt{G\_1 G\_2} \cos \Delta \Phi\\ \Delta \Phi &= \Phi\_1 - \Phi\_2 \end{aligned} \tag{6}$$

#### **3.2. QD-SOA–MZI-based XOR gate**

Fig. 3 depicts a schematic diagram of the all-optical QD-SOA-based MZI switch. It consists of symmetrical MZI where one QD-SOA is located in each arm of the interferometer [3,1,23]. A probe signal composed of continuous series of unit pulses at wavelength *λp* is inserted in the MZI and is split into two equal parts and travels separately along the identical QD-SOAs located in their path.

**Figure 6.** XOR gate based on QD-SOA [3].

The wavelength separation between *λS* and *λp* should be less than the homogeneous broaden‐ ing of the single QD gain to ensure effective cross-gain modulation. The data pulse is at least an order of magnitude stronger than the incoming pulse so that only the input pulse can alter QD-SOAs'opticalproperties.Inthe caseofinputdataA=B=0,the travelingprobe signalthrough the two arms of the SOA acquires a phase difference of *π* when it recombines at the output, and so the outputis "0"due to thedestructive interference.Besides,inthe case ofA=1,B=0,the signal traveling through the arm acquires a phase change due to the presence of XPM between the pulse train and the signal. The signal traveling through the lower arm does not have this additional phase change which results in an output "1". The same result occurs for A=0, and B=1. When A=1 and B=1, the phase changes for the signal traveling through both arms will be equal, and the output is "0". The XOR output intensity can be expressed as [27–29, 32]

$$P\_{\rm{MCR}} = P\_{\rm{probe}}(k\_1 k\_2 \mathcal{G}\_A(t) + (1 - k\_1)(1 - k\_2)\mathcal{G}\_\mathbb{B}(t) - 2\sqrt{k\_1 k\_2 (1 - k\_1)(1 - k\_2)\mathcal{G}\_A(t)\mathcal{G}\_\mathbb{B}(t)}) \times \cos\left[\boldsymbol{\wp}\_A - \boldsymbol{\wp}\_\mathbb{B}\right] \tag{7}$$

Where *GA*(*t*) and *GB* (*t*) are defined as integrated gain of QD-SOA, *φ*1(*t*), *φ*2(*t*), are nonlinear phase shifts, *k*1, *k*<sup>2</sup> are the coupling coefficients of the couplers *C*1 and *C*2, respectively, and, in this work, are equally set to 0.5 for simplicity. We present some recommendations for the parameter designs for practical QD-SOA devices.


#### **3.3. QD-SOA–MZI-based AND gate**

**Figure 5.** Schematic of SOA-incorporated MZI structure [26].

92 Some Advanced Functionalities of Optical Amplifiers

*D*

*D*

**3.2. QD-SOA–MZI-based XOR gate**

**Figure 6.** XOR gate based on QD-SOA [3].

located in their path.

Denoting the input and output signal powers with PA1, PA2, PD1, and PD2 and assuming an

cos

cos

== + - DF (5)

(6)

ideal 3 dB coupler (sin*ki* = 2 / 2, cos*ki* = 2 / 2), the transfer function reduces to

1 1 2 12

2 1 2 12

*<sup>P</sup> <sup>H</sup> G G GG*

*<sup>P</sup> H G G GG*

111

442

111

== + + DF

Fig. 3 depicts a schematic diagram of the all-optical QD-SOA-based MZI switch. It consists of symmetrical MZI where one QD-SOA is located in each arm of the interferometer [3,1,23]. A probe signal composed of continuous series of unit pulses at wavelength *λp* is inserted in the MZI and is split into two equal parts and travels separately along the identical QD-SOAs

The wavelength separation between *λS* and *λp* should be less than the homogeneous broaden‐ ing of the single QD gain to ensure effective cross-gain modulation. The data pulse is at least

442

1

*D*

*A*

*P*

1

2

*D*

*A*

*P*

DF = F - F

1 1 2

> Logic AND operation is another important Boolean function which corresponds to the sampling of one signal with another. The AND gate is obtained using cross-phase modulation of two input signals in SOAs located in the two arms of a Mach–Zehnder interferometer built using SOAs. The principle of logic AND using the MZI involves coupling the two input signals

into ports 1 and 3 of the SOA–MZI configuration. If the operation is performed on the positive slope in the transfer function of interferometer, the input signal 2 will sample input signal 1. Therefore, the selective switching of the data pulses at *λ*AND occurs exclusively during the mark of signal 1, which yields the logical AND operation. The structural parameters are similar to the parameters considered for XOR gate [26, 41–44].

**Figure 7.** Configuration of a SOA–MZI structure for AND operation [26].

#### **3.4. Metrics characterizing the quality of switching in this chapter**

The quality factor *Q* =20×log((*P*<sup>1</sup> −*P*0) / (*σ*<sup>1</sup> + *σ*0)) where *P*1, *P*0,and *σ*1, and *σ*0 are the mean and the standard deviation of the peak power of the output's '1's and '0's, respectively. *Q* value is sensitive to the input pulse width and increasing the pulse width decreases the *Q* factor because of the overlapping of two neighboring pulses. The quality factor (*Q*) is dependent on Γ. Multilayer QD structures are considered as a technique to increase the modal gain due to increase in the Γ parameter and therefore reducing the current threshold. The extinction ratio 1

(ER) is defined as ER=10log( *<sup>P</sup>*MIN *P*MAX <sup>0</sup> ) where *P*<sup>1</sup> MIN and *P*<sup>0</sup> MAX are the minimum and maximum

values of the peak power of high state and low state, respectively.

## **4. Rate equation**

The typical structure of the quantum-dot SOA is illustrated in Fig. 5. The physics of operation of SOA includes the current injection into the active layer having quantum dots, and therefore the input optical signals are amplified through the stimulated emission or the use of the optical nonlinearity by the quantum dots for this processing. Fig. 5 also shows the cross-sectional and plan-view images of self-assembled InGaAs quantum dots as a typical example of quantumdot crystals. As it is clear, self-assembled InGaAs quantum dots on GaAs substrates and their application to semiconductor lasers have been studied since the early 1990s. They are nano‐ sized semiconductor islands with a wetting layer grown via the Stranski–Krastanov mode under highly mismatched epitaxy, where the electron energy states are completely quantized due to the three-dimensional quantum confinement. Quantum-dot SOAs are novel optical devices using self-assembled quantum dots [34].

**Figure 8.** Structure of the quantum-dot SOA with cross-sectional and plan-view images of self-assembled InGaAs quantum dots. [34]

**Figure 9.** Band diagram of the QD structure with related energy levels [3].

into ports 1 and 3 of the SOA–MZI configuration. If the operation is performed on the positive slope in the transfer function of interferometer, the input signal 2 will sample input signal 1. Therefore, the selective switching of the data pulses at *λ*AND occurs exclusively during the mark of signal 1, which yields the logical AND operation. The structural parameters are similar to

The quality factor *Q* =20×log((*P*<sup>1</sup> −*P*0) / (*σ*<sup>1</sup> + *σ*0)) where *P*1, *P*0,and *σ*1, and *σ*0 are the mean and the standard deviation of the peak power of the output's '1's and '0's, respectively. *Q* value is sensitive to the input pulse width and increasing the pulse width decreases the *Q* factor because of the overlapping of two neighboring pulses. The quality factor (*Q*) is dependent on Γ. Multilayer QD structures are considered as a technique to increase the modal gain due to increase in the Γ parameter and therefore reducing the current threshold. The extinction ratio

MIN and *P*<sup>0</sup>

The typical structure of the quantum-dot SOA is illustrated in Fig. 5. The physics of operation of SOA includes the current injection into the active layer having quantum dots, and therefore the input optical signals are amplified through the stimulated emission or the use of the optical nonlinearity by the quantum dots for this processing. Fig. 5 also shows the cross-sectional and plan-view images of self-assembled InGaAs quantum dots as a typical example of quantumdot crystals. As it is clear, self-assembled InGaAs quantum dots on GaAs substrates and their application to semiconductor lasers have been studied since the early 1990s. They are nano‐ sized semiconductor islands with a wetting layer grown via the Stranski–Krastanov mode under highly mismatched epitaxy, where the electron energy states are completely quantized due to the three-dimensional quantum confinement. Quantum-dot SOAs are novel optical

MAX are the minimum and maximum

the parameters considered for XOR gate [26, 41–44].

94 Some Advanced Functionalities of Optical Amplifiers

**Figure 7.** Configuration of a SOA–MZI structure for AND operation [26].

(ER) is defined as ER=10log( *<sup>P</sup>*MIN

devices using self-assembled quantum dots [34].

**4. Rate equation**

**3.4. Metrics characterizing the quality of switching in this chapter**

1 *P*MAX

values of the peak power of high state and low state, respectively.

<sup>0</sup> ) where *P*<sup>1</sup>

In the QD-SOA–MZI, optical signals propagate in an active medium with the gain determined by the rate equations for the electron transitions in QD-SOA between WL, ground state (GS) and ES. We have considered the two energy levels in the conduction band: GS and ES. The diagram of the energy levels and electron transitions in the QD conduction band is shown in Fig. 5. The stimulated and spontaneous radiative transitions occur from GS to the QD valence band. The system of the rate equations accounts for the following transitions:

**1.** The fast electron transitions from WL to ES with the relaxation time 5×1010 cm−2.


The balance between the WL and ES is determined by the shorter time of QDs filling. Carriers relax quickly from the ES level to the GS level, while the former serves as a carrier reservoir for the latter. In general, the radiative relaxation time depends on the bias current. However, it can be shown that for moderate values of the WL carrier density *Nw* =(10<sup>14</sup> <sup>−</sup>1015), this dependence can be neglected. The spontaneous radiative time in QDs *τ*1*<sup>R</sup>* remains large enough: *τ*1*<sup>R</sup>* ≥(0.4−0.5)ns [1]. In the case of the smaller signal detuning than the QD spectrum homogeneous broadening, the electron rate equations have the following forms [3,1,35,36]:

$$\frac{\partial S\_{\text{centrad}}(z,\tau)}{\partial z} = (-\alpha\_{\text{abs}})S\_{\text{centl}}\tag{8}$$

$$\frac{\partial \mathcal{S}\_{s\_{\text{sugar}}}(z,\tau)}{\partial z} = (\mathcal{g}(h\nu\_{s\_{\text{sugar}}}) - \alpha\_{\text{int}})\mathcal{S}\_{s\_{\text{sugar}}}\tag{9}$$

$$\frac{\partial N\_w(\boldsymbol{\varepsilon}, \boldsymbol{\tau})}{\partial \boldsymbol{\tau}} = \frac{J\_0}{eL\_w} - \frac{N\_w(1-h)}{\boldsymbol{\tau}\_{w2}} + \frac{N\_w h}{\boldsymbol{\tau}\_{2w}} - \frac{N\_w}{\boldsymbol{\tau}\_{wr}} \tag{10}$$

$$\begin{split} \frac{\partial f(\mathbf{z}, \boldsymbol{\tau})}{\partial \boldsymbol{\tau}} &= \frac{(1-f)h}{\boldsymbol{\tau}\_{21}} - \frac{f(1-h)}{\boldsymbol{\tau}\_{12}} - \frac{f^2}{\boldsymbol{\tau}\_{1R}} - \frac{g\_{\max,S}L}{N\_Q}(2f-1) \\ \times S\_{\text{res}}(\mathbf{z}, \boldsymbol{\tau}) \frac{c}{\sqrt{\xi\_r}} &- \frac{g\_{\max,P}L}{N\_Q}(2f-1) \times S\_{\text{syd}}(\mathbf{z}, \boldsymbol{\tau}) \frac{c}{\sqrt{\xi\_r}} \end{split} \tag{11}$$

$$\frac{\partial h(z,\tau)}{\partial \tau} = \frac{L\_w N\_w (1-h)}{N\_Q \tau\_{w2}} - \frac{N\_w L\_w h}{N\_Q \tau\_{2w}} - \frac{(1-f)h}{\tau\_{21}} + \frac{f(1-h)}{\tau\_{12}} + \frac{\alpha\_{\text{max}} L}{N\_Q} (1-2h) \times S\_{\text{control}}(z,\tau) \frac{c}{\sqrt{\xi\_r}} \tag{12}$$

where *e* is the electron charge and *J* is the injection current density and bias current equal to 50 mA. In addition, *τw*2 is the electron relaxation time from the WL to the ES, *τ*2*<sup>w</sup>* is the electron escape time from the ES to the WL, *τwR* is the spontaneous radiative lifetime in WL, *τ*21 is the electron relaxation time from the ES to the GS, *τ*12 is the electron escape time from the GS to the ES, and *τ*1*<sup>R</sup>* is the spontaneous radiative lifetime in the QD. *NQ* is the surface density of QDs where its typical value is 5×1010 cm−2 *α*, *Nw* is the electron density in the WL, *L <sup>w</sup>* is the effective thickness of the active layer, *ξ<sup>r</sup>* is the SOA material permittivity and is the velocity of light in free space. The last term in eq. (5) and last two terms in eq. (6) demonstrate the absorption of CP and stimulated emission in the conduction band ground state (CBGS), respectively. For simplicity, we presume an ideal facet reflectivity and neglect the amplified spontaneous emission. The time dependence of the integral QD-SOA gain and pulse-phase shift can be expressed as *G*(*τ*)=exp(*∫* 0 *L g*(*z* ' , *τ*)*dz*) and *ϕ*(*τ*)= −*α* / 2(*∫* 0 *L g*(*z* ' , *τ*)dz), respectively, where *α* is the linewidth enhancement factor (LEF). It has been discussed in several articles that linewidth enhancement factor may vary in a large interval from the experimentally measured value of 0.1 up to giant values of 60 in QDs. The set of Eqs. 8–12 with the defined initial conditions cannot be solved in a closed form but it can only be solved numerically. For this purpose, the optical pulses and the SOAs have been divided in many small segments in time and distance, respectively, and solutions have been obtained stepwise both in time and space for the temporal gain and phase changes experienced by the clock pulses in the two arms of the interferometer. These are required to calculate the characteristics of the switched-out clock pulses at the transmission and reflection port of the interferometer, expressed by the equations [25,27,28,38]

$$P\_{\rm xCR} = P\_{\rm probe} \left( k\_1 k\_2 G\_A(t) + (1 - k\_1)(1 - k\_2) G\_\mathbf{z}(t) - 2 \sqrt{k\_1 k\_2 (1 - k\_1)(1 - k\_2) G\_A(t) G\_\mathbf{z}(t)} \right) \times \cos\left[\varphi\_A - \varphi\_\mathbf{z}\right] \tag{13}$$

$$P\_{\rm cross} = P\_{\rm probe} \left\{ k\_1 k\_2 G\_A(t) + (1 - k\_1)(1 - k\_2) G\_B(t) + 2 \sqrt{k\_1 k\_2 (1 - k\_1)(1 - k\_2) G\_A(t) G\_B(t)} \right\} \times \cos\left[\varphi\_A - \varphi\_B\right] \tag{14}$$

*P*XOR, *P*CROSS are the transmission and reflection functions.

**2.** The fast electron transitions between ES and GS with the relaxation time from ES to GS

**3.** The slow transitions of electrons escaping from ES back to WL with the electron escape

The balance between the WL and ES is determined by the shorter time of QDs filling. Carriers relax quickly from the ES level to the GS level, while the former serves as a carrier reservoir for the latter. In general, the radiative relaxation time depends on the bias current. However, it can be shown that for moderate values of the WL carrier density *Nw* =(10<sup>14</sup> <sup>−</sup>1015), this dependence can be neglected. The spontaneous radiative time in QDs *τ*1*<sup>R</sup>* remains large enough: *τ*1*<sup>R</sup>* ≥(0.4−0.5)ns [1]. In the case of the smaller signal detuning than the QD spectrum homogeneous broadening, the electron rate equations have the following forms [3,1,35,36]:

control abs

signal,prob signal,prob int

a

2 2 wr

*R Q*

max,

*S*

 t

max

a

 x

control

t

x

(11)

 tt

2

*r Q r*

*Q w Q w Q r*

¶ (12)

 t

where *e* is the electron charge and *J* is the injection current density and bias current equal to 50 mA. In addition, *τw*2 is the electron relaxation time from the WL to the ES, *τ*2*<sup>w</sup>* is the electron escape time from the ES to the WL, *τwR* is the spontaneous radiative lifetime in WL, *τ*21 is the electron relaxation time from the ES to the GS, *τ*12 is the electron escape time from the GS to the ES, and *τ*1*<sup>R</sup>* is the spontaneous radiative lifetime in the QD. *NQ* is the surface density of QDs where its typical value is 5×1010 cm−2 *α*, *Nw* is the electron density in the WL, *L <sup>w</sup>* is the effective thickness of the active layer, *ξ<sup>r</sup>* is the SOA material permittivity and is the velocity of light in free space. The last term in eq. (5) and last two terms in eq. (6) demonstrate the

*S*

¶ = - ¶ (8)

<sup>=</sup> - ¶ (9)

¶ (10)

(,) ( )

a

(( ) )

(,) (1 ) *<sup>w</sup> w ww ww w*

=- + -

 t

( , ) (1 ) (1 ) <sup>1</sup>

¶ -- = - - - (2 - ) ¶

(,) 1 (,)

*c c g L S z f Sz <sup>N</sup>*

(,) (1 ) (1 ) (1 ) (1 2 ) ( , ) *w w w w*

*h z LN h NLh fh f h <sup>L</sup> <sup>c</sup> hS z*

*N z J N h Nh N*

*g hw S*

t

*τ*<sup>21</sup> =0.16 ps and the relaxation time from GS to ES *τ*<sup>21</sup> =1.2 ps.

control

(,)

t

signal,prob

¶

t

t

t

 t

t

t

*S z*

*z*

t

t

 t

x

 tt

*S z*

*z*

0

Prob signal

´ - - (2 - ) ´

2 2 21 12

*N N N*

¶ - - - = - - + + -´

21 12 1 max,

 tt

*fz fh f h f g L <sup>f</sup> <sup>N</sup>*

*P*

*eL*

¶ -

time *τ*2*<sup>w</sup>* =1 ns.

96 Some Advanced Functionalities of Optical Amplifiers

$$\mathcal{g}(hw\_{\text{s/gal,prob}}) = \mathcal{g}\_{\text{max}}(\mathcal{D}f - 1) \tag{15}$$

$$\alpha\_{\rm abs} = \alpha\_{\rm max} (1 - 2h) + \alpha\_{\rm int}, S \text{ (} z, \tau \text{)} = \frac{p(z, \tau)}{A\_{\rm eff} V\_{\rm g} h \upsilon}, p(z, t) = \sum\_{l=1}^{n} P\_{\rm max} \exp(\frac{-4 \ln 2 (t - nT)^2}{\tau^2\_{\rm FWHM}}) \tag{16}$$

The wavelengths of signal, probe, and CP are considered to be: *λ<sup>s</sup>* =1.56 μm, *λ<sup>p</sup>* =1.53 μm and *λ<sup>c</sup>* =1.31 μm. For the following structure parameters [3,35,36, 37,39]:

$$\begin{aligned} \text{g}\_{\text{max}} &= 11.5 \,\text{cm}^{-1}, \alpha\_{\text{int}} = 3 \,\text{cm}^{-1}, N\_{\text{Q}} = 5 \times 10^{10} \,\text{cm}^{-2}, L\_w = 0.25 \,\mu\text{m}, \\ \text{g}\_{\text{max}} &= 3 \,\text{ps}, \tau\_{2w} = 1 \,\text{ns}, \tau\_{w\text{R}} = 0.2 \,\text{ns}, \tau\_{21} = 0.16 \,\text{ps}, \tau\_{12} = 1.2 \,\text{ps}, \tau\_{1\text{R}} = 0.4 \,\text{ns}, \\ \text{W} &= 10 \,\text{\mu m}, \Gamma = 3 \times 10^{-2}, \alpha\_{\text{max}} = 10 \,\text{cm}^{-1}, \alpha\_{\text{LF}} = 1, \text{W} = 10 \,\mu\text{m}, \Gamma = 3 \times 10^{-2}, \alpha\_{\text{max}} = 10 \,\text{cm}^{-1} \end{aligned}$$

#### **5. Simulation results and discussion**

Achieving high-speed signal processing, as mentioned earlier, depends strongly on WL to ES and ES to GS relaxation times. However, *τ*21 is not a limiting factor in the operation of con‐ ventional QD-SOAs for 200 Gb/s. According to the reported results [1] pertaining to this case, *τ*21 is 160 fs because of longer WL to ES relaxation time. However, this parameter can be important in achieving high-speed operation in the proposed approach as a higher limit. Increasing the *τ*<sup>21</sup> =0.4 ps relaxation time and consequence *τ*12(*τ*<sup>12</sup> =*τ*21exp (*E*<sup>2</sup> −*E*1) / *KBT* ) will decrease the quality factor for XOR and AND output. *KB* is the Boltzmann constant, *T* is the absolute temperature, and *E*2−*E*1 is the energy separation between the ES and GS. Numerically calculated results are illustrated in the following figures:

Fig. 10‐a. Input waveforms of all‐optical half‐adder with input data stream A. Fig.10‐b. Input waveforms of all‐optical half‐adder with input data stream B. **Figure 10.** (a). Input waveforms of all-optical half-adder with input data stream A. (b). Input waveforms of all-optical half-adder with input data stream B.

Absorption of the CP will populate the conduction band excitation state (CBES) and hence the recovery process will accelerate, and the recovery process is much faster compared with the state where no CP is applied in the same input power and injected current. Absorption of the CP will populate the conduction band excitation state (CBES) and hence the recovery process will accelerate, and the recovery process is much faster compared with the state where no CP is applied in the same input power and injected current.

At 2 Tb/s bit sequence, the population variation cannot reach the final population value but still varies with relatively high amplitude.

**h(t)** At both bit rates of 1 and 2 Tb/s, the oscillation of ES and GS completely follows the input signal variation. The simulated output waveforms are shown in Fig. 13 (a) and (b), respectively, where Fig. 13 (a) illustrates the output sum-bit and 13 (b) shows output carry-bit. Fig. 13 (c) shows that the value of ER is decreased as this relaxation time is increased to 0.4 ps. With

Fig. 11‐a. Electron state occupation probabilities of ES, *h*(*t*) with and without CP at 1 Tb/s. The bias current is 50 mA, input signal, CP, and probe signal powers are 200 UW, 250 UW, and 2 UW, respectively.

increase in the bit-rate value from 200 Gb/s, the ER=10log( *P*MIN 1 *P*MAX <sup>0</sup> ) is decreased as well.

ventional QD-SOAs for 200 Gb/s. According to the reported results [1] pertaining to this case, *τ*21 is 160 fs because of longer WL to ES relaxation time. However, this parameter can be important in achieving high-speed operation in the proposed approach as a higher limit. Increasing the *τ*<sup>21</sup> =0.4 ps relaxation time and consequence *τ*12(*τ*<sup>12</sup> =*τ*21exp (*E*<sup>2</sup> −*E*1) / *KBT* ) will decrease the quality factor for XOR and AND output. *KB* is the Boltzmann constant, *T* is the absolute temperature, and *E*2−*E*1 is the energy separation between the ES and GS. Numerically

(a)

(b) Fig. 10‐a. Input waveforms of all‐optical half‐adder with input data stream A. Fig.10‐b. Input waveforms of all‐optical half‐adder with input data stream B.

Absorption of the CP will populate the conduction band excitation state (CBES) and hence the recovery process will accelerate, and the recovery process is much faster compared with the state where no CP is applied in the same

Absorption of the CP will populate the conduction band excitation state (CBES) and hence the recovery process will accelerate, and the recovery process is much faster compared with the

At 2 Tb/s bit sequence, the population variation cannot reach the final population value but

At both bit rates of 1 and 2 Tb/s, the oscillation of ES and GS completely follows the input signal variation. The simulated output waveforms are shown in Fig. 13 (a) and (b), respectively, where Fig. 13 (a) illustrates the output sum-bit and 13 (b) shows output carry-bit. Fig. 13 (c) shows that the value of ER is decreased as this relaxation time is increased to 0.4 ps. With

state where no CP is applied in the same input power and injected current.

**Figure 10.** (a). Input waveforms of all-optical half-adder with input data stream A. (b). Input waveforms of all-optical

Fig. 11‐a. Electron state occupation probabilities of ES, *h*(*t*) with and without CP at 1 Tb/s. The bias current is 50 mA, input signal, CP, and probe signal powers are 200 UW, 250 UW, and 2 UW, respectively.

*P*MIN 1 *P*MAX

<sup>0</sup> ) is decreased as well.

calculated results are illustrated in the following figures:

98 Some Advanced Functionalities of Optical Amplifiers

input power and injected current.

still varies with relatively high amplitude.

increase in the bit-rate value from 200 Gb/s, the ER=10log(

half-adder with input data stream B.

**h(t)**

**Figure 11.** (a). Electron state occupation probabilities of ES, *h*(*t*) with and without CP at 1 Tb/s. The bias current is 50 mA, input signal, CP, and probe signal powers are 200 UW, 250 UW, and 2 UW, respectively. (b). Electron state occu‐ pation probabilities of ES, *f*(*t*) with and without CP at 1 Tb/s. The bias current is 50 mA, input signal, CP, and probe signal powers are200 UW, 250 UW, and 2 UW, respectively.

**Figure 12.** (a). Electron state occupation probabilities of ES, *h(t)* with and without CP at 2 Tb/s. The bias current is 50 mA, input signal, CP, and probe signal powers are 200 UW, 250 UW, and 2 UW, respectively. (b). Electron state occu‐ pation probabilities of ES, *f*(*t*) with and without CP at 2 Tb/s. The bias current is 50 mA, input signal, CP, and probe signal powers are 200 UW, 250 UW, and 2 UW, respectively.

**Figure 13.** (a). Output waveforms of all-optical half-adders, where output sum-bit at 1Tb/s, the results describe that the pattern effect is negligible at 1 Tb/s. (b). Output waveforms of all-optical half-adders, where output carry-bit at 1 Tb/s.

**Figure 14.** Output waveforms of all-optical half-adder, where output carry-bit at 1 Tb/s with *τ*<sup>21</sup> =0.4*ps*.

The ER is changed with the change in the electron relaxation time during electron transport from the GS to the ES. Here, the value of ER is decreased in an exponential-like manner as this relaxation time is increased. Therefore, the transition time between GS and ES must be kept low as fast as possible. The device gain is determined by the carrier density of the QD ground state. As the carrier remains longer in GS before going to the ES level,,the transition rate from the ES to GS accelerates and the ES population rapidly decreases. Decrease in the ES population reduces the gain magnitude. The WL serves as the only recipient of the pump current, while QD's excited state serves as a carrier reservoir for the GS with ultrafast carrier relaxation to the latter, and their carrier density and transition rates can affect the device gain. Therefore, the slow electron transition between ES and WL occurs, which leads to the slowing down of the cross-gain modulation process. Hence, the extinction ratio is decreased.

**Figure 15.** Variation of ER with electron relaxation time from the ES to the GS for input bit sequences at 1 Tb/s.

**Figure 16.** Quality factor with electron relaxation time from the ES to the GS for input bit sequences at 1 Tb/s.

**Figure 13.** (a). Output waveforms of all-optical half-adders, where output sum-bit at 1Tb/s, the results describe that the pattern effect is negligible at 1 Tb/s. (b). Output waveforms of all-optical half-adders, where output carry-bit at 1 Tb/s.

100 Some Advanced Functionalities of Optical Amplifiers

**Figure 14.** Output waveforms of all-optical half-adder, where output carry-bit at 1 Tb/s with *τ*<sup>21</sup> =0.4*ps*.

the cross-gain modulation process. Hence, the extinction ratio is decreased.

The ER is changed with the change in the electron relaxation time during electron transport from the GS to the ES. Here, the value of ER is decreased in an exponential-like manner as this relaxation time is increased. Therefore, the transition time between GS and ES must be kept low as fast as possible. The device gain is determined by the carrier density of the QD ground state. As the carrier remains longer in GS before going to the ES level,,the transition rate from the ES to GS accelerates and the ES population rapidly decreases. Decrease in the ES population reduces the gain magnitude. The WL serves as the only recipient of the pump current, while QD's excited state serves as a carrier reservoir for the GS with ultrafast carrier relaxation to the latter, and their carrier density and transition rates can affect the device gain. Therefore, the slow electron transition between ES and WL occurs, which leads to the slowing down of

Fig. 17 shows that the ER is very sensitive to the variations of the electron relaxation time from the ES to the GS since the slope of the curve is decreased in an exponential-like manner as this relaxation time is increased, finally becoming smoother near the left edge of the diagram. So the transition time between ES and GS must be kept below this limit and ideally be as fast as possible.

**Figure 17.** Variation of ER with electron relaxation time from the GS to the ES for input bit sequences at 1 Tb/s.

Fig. 18 illustrates the effect of sum with peak power of the input data signals on the ER. The characteristic of the curve is that the ER increases with power.

Fig. 19 illustrates the effect of carry with peak power of the input data signals on the ER. The characteristic of the curve is that the ER is increased with power up to a certain value after that it is decreased.

**Figure 18.** Variation of extinction ratio (ER) with peak data power for sum-bit, keeping other parameters fixed.

**Figure 19.** Variation of extinction ratio (ER) with peak data power for carry bit, keeping other parameters fixed.

**Figure 20.** (a). Output waveforms of all-optical half-adder, sum-bit at 2 Tb/s. (b). Output waveforms of all-optical halfadder, carry-bit at 2 Tb/s.

The quality factor and ER are decreased with bit rate since at 2 Tb/s bit sequence, the population variation cannot reach the final population value but still varies with relatively high amplitude.

**Figure 21.** Variation of ER with electron relaxation time from the ES to the GS for input bit sequences at 2 Tb/s.

**Figure 22.** Quality factor with electron relaxation time from the ES to the GS for input bit sequences at 2 Tb/s.

#### **6. Conclusion**

The quality factor and ER are decreased with bit rate since at 2 Tb/s bit sequence, the population variation cannot reach the final population value but still varies with relatively high amplitude.

**Figure 20.** (a). Output waveforms of all-optical half-adder, sum-bit at 2 Tb/s. (b). Output waveforms of all-optical half-

adder, carry-bit at 2 Tb/s.

**Figure 18.** Variation of extinction ratio (ER) with peak data power for sum-bit, keeping other parameters fixed.

102 Some Advanced Functionalities of Optical Amplifiers

**Figure 19.** Variation of extinction ratio (ER) with peak data power for carry bit, keeping other parameters fixed.

A novel model of ultrafast all-optical half-adder using two QD-SOAs-based Mach–Zehnder interferometer was theoretically investigated and demonstrated. Numerically simulated results confirming the described method are also given in this chapter. The variation of ES to GS relaxation time on the ER and *Q*-factor at the output has been thoroughly investigated. We enhanced the bit rate, *Q* factor, and extinction ratio parameters for the half-adder. We intro‐ duced theoretical approaches to compensate the slow-carrier transition relaxation time from WL to ES (using XGM effect) which is the main limit to achieve higher speeds in QD-SOAs. It is concluded that the proposed approach accelerates the recovery process of the SOA. Apply‐ ing a CP to the two-energy level QD at certain times enables the QD-SOA–MZI-based halfadder to operate under 1 and 2 Tb/s input bit sequences. This capability of control-pulseassisted QD-SOA is promising for ultrahigh-speed all-optical logic gates, all-optical switching, and processing. The model can be extended for studying more complex all-optical circuits of enhanced functionality in which this proposed circuit developed in this paper may be assumed as the basic building block.

## **Author details**

Khalil Safari1 , Ali Rostami1,2\*, Ghasem Rostami2 and Mahboubed Dolatyari2

\*Address all correspondence to: rostami@tabrizu.ac.ir

1 Photonics and Nanocrystal Research Lab. (PNRL), Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran

2 SP-EPT Labs., ASEPE Company, Industrial Park of Advanced Technologies, Tabriz, Iran

## **References**


ferometer switches for demultiplexing to 10 and 40 Gbit/s," *Opt. Commun.,* vol. 189, no. 4–6, pp. 241–249, 2001.

[10] W. Hong, D. Huang, and G. Zhu, "Switching window of an SOA loop mirror with SOA sped-up by a CW assist light at transparency wavelength," *Opt. Commun.,* vol. 238, no. 1–3, pp. 151–156, 2004.

**Author details**

104 Some Advanced Functionalities of Optical Amplifiers

, Ali Rostami1,2\*, Ghasem Rostami2

1 Photonics and Nanocrystal Research Lab. (PNRL), Faculty of Electrical and Computer

2 SP-EPT Labs., ASEPE Company, Industrial Park of Advanced Technologies, Tabriz, Iran

[1] Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, "Ultrafast all-optical processor based on quantum-dot semiconductor optical amplifiers," *IEEE J. Quantum Electron.*, vol.

[2] B. Dai, S. Shimizu, X. Wang, and N. Wada, "Simultaneous all-Optical half-adder and half-subtracter based on two semiconductor optical amplifiers," *IEEE Photonics Tech‐*

[3] D. K. Gayen, D. Kumar, and T. Chattopadhyay, "Designing of optimized all-optical half adder circuit using single quantum-dot semiconductor optical amplifier assisted Mach-Zehnder interferometer," *J. Lightwave Technol.,* vol. 31, no. 12, pp. 2029–2035,

[4] A. Rostami, H. B. A. Nejad, R. M. Qartavol, and H. R. Saghai, "Tb/s optical logic gates based on quantum-dot semiconductor optical amplifiers," *IEEE J. Quantum*

[5] L. M. Sa, H. Silva, P. Andre, and R. Nogueira, "Simulation performance of all-optical logic gate XOR at 40 Gbit/s using quantum-dot SOAs," *2011 IEEE EUROCON – Inter‐*

[6] H. L. Minh, F. Z. Ghassemlooy, and W. P. Ng, "All-optical flip-flop based on a sym‐ metric Mach-Zehnder switch with a feedback loop and multiple forward set/reset

[7] H. Sun, Q. Wang, H. Dong, and N. K. Dutta, "XOR performance of a quantum dot semiconductor optical amplifier based Mach-Zehnder interferometer," *Opt. Exp.*, vol.

[8] R. Clavero, F. Ramos, J. M. Martinez, and J. Marti, "All-optical flip-flop based on a single SOA-MZI," *IEEE Photonics Technol. Lett.*, vol. 17, no. 4, pp. 843–845, 2005. [9] S. Diez, E. Hilliger, M. Kroh, C. Schmidt, C. Schubert, H. G. Weber, L. Occhi, L. Schares, G. Guekos, and L. K. Oxenloewe, "Optimization of SOA based Sagnac inter‐

\*Address all correspondence to: rostami@tabrizu.ac.ir

Engineering, University of Tabriz, Tabriz, Iran

45, no. 1, pp. 34–41, 2009.

*Electron.*, vol. 46, no. 3, pp. 354–360, 2010.

13, no. 6, pp. 1892–1899, 2005.

*national Conference on Computer as a Tool*, 2011.

signals," *Opt. Eng*., vol. 46, no. 4, pp. 40501–40503, 2007

*nol*. Lett., 2012.

2013.

and Mahboubed Dolatyari2

Khalil Safari1

**References**


[34] E. Dimitriadou and K. E. Zoiros, "On the feasibility of ultrafast all-optical NAND gate using single quantum-dot semiconductor optical amplifier-based Mach-Zehnder interferometer," *Opt. Laser Technol.*, vol. 44, no. 6, pp. 1971–1981, 2012.

[22] S. H. Kim, J. H. Kim, J. W. Choi, C.W. Son, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, and S. H. Kim, "All-optical half adder using cross-gain modulation in semiconductor

[23] P. L. Li, D. X. Huang, X. L. Zhang, and G. X. Zhu, "Ultrahigh speed all-optical half adder based on four-wave mixing in semiconductor optical amplifier," *Opt. Exp.*, vol.

[24] P. Phongsanam, S. Mitatha, C. Teeka, and P. P. Yupapin, "All optical half adder/ subtractor using dark-bright soliton conversion control," *Microw. Opt. Technol. Lett*.,

[25] J. W. M. Menezes, W. B. Fraga, A. C. Ferreira, G. F. Guimaraes, A. F. G. F. Filho, C. S. Sobrinho, and A. S. B. Sombra, "All-optical half adder using all-optical XOR and AND gates for optical generation of "Sum" and "Carry"," *Fiber Integr. Opt.*, vol. 29,

[26] R. P. Schreieck, M. H. Kwakernaak, H. Jackel, and H. Melchior, "All-optical switch‐ ing at multi-100-Gbit/s data rates with Mach-Zehnder interferometer switches," *IEEE*

[27] A. Kumar, S. Kumar, and S. K. Raghuwanshi, "Implementation of full-adder and fullsubtractor based on electro-optic effect in Mach–Zehnder interferometers," *Opt. Com‐*

[29] S. Nakamura, Y. Ueno, K. Tajima, J. Sasaki, T. Sugimoto, T. Kato, T. Shimoda, M. Itoh, H. Hatakeyama, T. Tamanuki, and T. Sasaki, "Demultiplexing of 168-Gb/s data pulses width a hybrid-integrated symmetric Mach-Zehnder all-optical switch," *IEEE*

[30] D. K. Gayen, A. Bhattachryya, T. Chattopadhyay, and J. N. Roy, "Ultrafast all-optical half adder using quantum-dot semiconductor optical amplifier based Mach-Zehnder

[31] [31]A. Rostami, H. Baghban, R. Maram, "Nanostructure Semiconductor Optical Am‐

[32] J. Y. Kim, J. M. Kang, T. Y. Kim, and S. K. Han, "All-optical multiple logic gates with XOR, NOR, OR, and NAND functions using parallel SOA-MZI structures: Theory

[33] H. Sun, Q. Wang, H. Dong, and N. K. Dutta, "XOR performance of a quantum-dot semiconductor optical amplifier based Mach-Zender interferometer," *Opt. Exp.*, vol.

interferometer," *J. Lightwave Technol*., vol. 30, no. 21, pp. 3387-3393, 2012.

and experiment," *J. Lightwave Technol*., vol. 24, no. 9, pp. 3392–3399, 2006.

*J. Quantum Electron*., vol. 38, no. 8, pp. 1053–1061, 2002.

[28] A. Rostami, "Applications and Functionalities, "*Eng. Mater*, 2011.

*Photon. Technol. Lett.*, vol. 12, no. 5, pp. 425–427, 2000.

plifiers," Berlin, Germany: Springer-Verlag, 2011.

13, no. 6, pp. 1892–1899, 2005.

optical amplifiers," *Opt. Exp.,* vol. 14, no. 22, pp. 10693–10698, 2006.

14, no. 24, pp. 11839–11847, 2006.

106 Some Advanced Functionalities of Optical Amplifiers

vol. 53, no. 7, pp. 1541–1544, 2011.

no. 4, pp. 254–271, 2010.

*mun*., 2014.


**Semiconductor Optical Amplifier (SOA)–Based Amplification of Intensity-Modulated Optical Pulses — Deterministic Timing Jitter and Pulse Peak Power Equalization Analysis**

T. Alexoudi, G.T. Kanellos, S. Dris, D. Kalavrouziotis, P. Bakopoulos, A. Miliou and N. Pleros

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61712

#### **Abstract**

During the last few years, large-scale efforts towards realizing high-photonic inte‐ gration densities have put SOAs in the spotlight once again. Hence, the need to de‐ velop a complete framework for SOA-induced signal distortion to accurately evaluate a system's performance has now become evident. To cope with this de‐ mand, we present a detailed theoretical and experimental investigation of the deter‐ ministic timing jitter and the pulse peak power equalization of SOA-amplified intensity-modulated optical pulses. The deterministic timing jitter model relies on the pulse mean arrival time estimation and its analytic formula reveals an approxi‐ mate linear relationship between the deterministic timing jitter and the logarithmic values of intensity modulation when the SOA gain recovery time is faster than the pulse period. The theoretical analysis also arrives at an analytic expression for the intensity modulation reduction (IMR), which clearly elucidates the pulse peak pow‐ er equalization mechanism of SOA. The IMR analysis shows that the output intensi‐ ty modulation depth is linearly related to the respective input modulation depth of the optical pulses when the gain recovery time is faster than the pulse period. This novel theoretical platform provides a qualitative and quantitative insight into the SOA performance in case of intensity-modulated optical pulses.

**Keywords:** Deterministic timing jitter, Pulse peak power equalization, Intensity modula‐ tion reduction, Semiconductor optical amplifier, Modulation depth index

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

Semiconductor optical amplifiers (SOAs) have long been the subject of considerable research interest, mainly exploiting their nonlinear properties to provide fast all-optical signal proc‐ essing [1], such as high-speed wavelength conversion (WC) [2, 3], bitwise logic operations [4-6], and signal regeneration [7]. The broad-scale efforts towards realizing high photonic integration densities have, however, put the use of SOAs as amplification elements in the spotlight once again, since any alternative integrated amplifier competitor [8] lags far behind in terms of integration maturity. SOAs currently emerge as the preeminent on-chip amplifier solution and their reintroduction in the toolbox of the optical network designer is now evident in many key network subsystems. As a result, multiple demonstrations of SOAs performing as pure amplifier stages [9, 10] or as ON-OFF gating elements [11], where amplification occurs in the ON state, have been presented. Their ubiquitous use spans diverse network segments, enabling leading edge applications that extend from metro [11] to access network environ‐ ments [10] and to on-chip or on-board datacom systems [9].

A concerted research effort on SOA-based devices, spanning the last 20 years, has unraveled most of their underlying amplification secrets, addressing a variety of linear and nonlinear phenomena and their impact on a system's performance [12]. Pulse-shaped asymmetry owing to SOA saturation effects, for example, has been one of the key findings and has been exten‐ sively studied for the past years [13]. However, it was only recently that a novel theoretical analysis correlated this behavior to SOA-induced deterministic timing jitter that optical pulses experience during the amplification process, also suggesting an analytic mathematical formula for its accurate estimation [14, 15]. On the other hand, amplitude modulation phenomena for SOA in-line amplification have been theoretically studied [16, 17] but the pulse peak power equalization properties of SOAs, although experimentally utilized in many cases [18–20], have never been expressed in an analytical form that would allow a straightforward estimation for any case of input signal. So far, the pulse peak power equalization properties of SOAs have been theoretically and experimentally investigated only for the SOA-based interferometric switches [20]. As a result, the proposed theoretical model cannot be applied for single SOA inline amplification cases, since it relies on cross-phase modulation (XPM) phenomena that take place in SOA-based interferometric devices. Although research efforts have shed plenty of light on the SOA-based amplification process during the last few years, a complete framework for SOA-induced signal distortion in case of intensity-modulated optical pulses, including both deterministic timing jitter and the intensity modulation reduction analysis, is still missing.

In order to fill the current gap in the system's performance assessment, we present here a holistic theoretical analysis for the SOA-based amplification process along with its experi‐ mental verification when intensity-modulated optical pulses are inserted into the amplifier. The aim of this chapter is to provide a systematic methodology on the origin, nature and quantification of SOA-induced deterministic timing jitter and pulse peak power equalization that intensity-modulated optical pulses experience during the amplification process. At first, an analytic formula for the pulse mean arrival time at the SOA exit is derived, providing a comprehensive picture of jitter origin and allowing for reliable estimation of the deterministic jitter induced during the SOA amplification. The theoretical analysis continues with an analytical mathematical expression of intensity modulation reduction induced by SOA amplification. More specifically, the output-versus-input modulation depth of the amplifier is examined for several saturation levels to thoroughly investigate the pulse peak power equalization capabilities of the SOA. The theoretical models are also experimentally verified with the obtained results proving good agreement between theory and experimental obser‐ vations, in both cases. Moreover, the deterministic timing jitter analysis reveals an approximate linear relationship between jitter values and the logarithm of pulse peak power modulation. Both experimental and theoretical results show that deterministic timing jitter minimization can be achieved by operating the SOA in the strongly saturated region. On the other hand, pulse peak power equalization analysis indicates a linear dependence between the output and input modulation depth indices. In that case, results show that the amplifier yields higher intensity modulation reduction values when it is operated in the saturation regime and for increased SOA gain levels.

In this perspective, the following sections of the chapter have been organized so as to introduce the concept and provide the analytical theoretical framework of the SOA-induced determin‐ istic timing jitter and the pulse peak power equalization properties for intensity-modulated optical pulses, as well as to describe the experimental setup along with the respective results obtained in each case and finally discuss potential extensions of the proposed theoretical models.

## **2. Concept and theoretical analysis**

**1. Introduction**

110 Some Advanced Functionalities of Optical Amplifiers

Semiconductor optical amplifiers (SOAs) have long been the subject of considerable research interest, mainly exploiting their nonlinear properties to provide fast all-optical signal proc‐ essing [1], such as high-speed wavelength conversion (WC) [2, 3], bitwise logic operations [4-6], and signal regeneration [7]. The broad-scale efforts towards realizing high photonic integration densities have, however, put the use of SOAs as amplification elements in the spotlight once again, since any alternative integrated amplifier competitor [8] lags far behind in terms of integration maturity. SOAs currently emerge as the preeminent on-chip amplifier solution and their reintroduction in the toolbox of the optical network designer is now evident in many key network subsystems. As a result, multiple demonstrations of SOAs performing as pure amplifier stages [9, 10] or as ON-OFF gating elements [11], where amplification occurs in the ON state, have been presented. Their ubiquitous use spans diverse network segments, enabling leading edge applications that extend from metro [11] to access network environ‐

A concerted research effort on SOA-based devices, spanning the last 20 years, has unraveled most of their underlying amplification secrets, addressing a variety of linear and nonlinear phenomena and their impact on a system's performance [12]. Pulse-shaped asymmetry owing to SOA saturation effects, for example, has been one of the key findings and has been exten‐ sively studied for the past years [13]. However, it was only recently that a novel theoretical analysis correlated this behavior to SOA-induced deterministic timing jitter that optical pulses experience during the amplification process, also suggesting an analytic mathematical formula for its accurate estimation [14, 15]. On the other hand, amplitude modulation phenomena for SOA in-line amplification have been theoretically studied [16, 17] but the pulse peak power equalization properties of SOAs, although experimentally utilized in many cases [18–20], have never been expressed in an analytical form that would allow a straightforward estimation for any case of input signal. So far, the pulse peak power equalization properties of SOAs have been theoretically and experimentally investigated only for the SOA-based interferometric switches [20]. As a result, the proposed theoretical model cannot be applied for single SOA inline amplification cases, since it relies on cross-phase modulation (XPM) phenomena that take place in SOA-based interferometric devices. Although research efforts have shed plenty of light on the SOA-based amplification process during the last few years, a complete framework for SOA-induced signal distortion in case of intensity-modulated optical pulses, including both deterministic timing jitter and the intensity modulation reduction analysis, is still missing.

In order to fill the current gap in the system's performance assessment, we present here a holistic theoretical analysis for the SOA-based amplification process along with its experi‐ mental verification when intensity-modulated optical pulses are inserted into the amplifier. The aim of this chapter is to provide a systematic methodology on the origin, nature and quantification of SOA-induced deterministic timing jitter and pulse peak power equalization that intensity-modulated optical pulses experience during the amplification process. At first, an analytic formula for the pulse mean arrival time at the SOA exit is derived, providing a comprehensive picture of jitter origin and allowing for reliable estimation of the deterministic

ments [10] and to on-chip or on-board datacom systems [9].

It is a well-known fact that intensity-modulated optical pulses will experience a pulse-shaped distortion and intensity modulation suppression when propagating through the SOA. The shift of the amplified pulse peak towards its rising edge owes to the higher gain that the leading edge of every incoming pulse experiences compared to the gain received by the trailing edge of the pulse [13]. This "center of gravity" deviation of the exiting optical pulse indicates a subsequent deviation of the mean pulse arrival time *T*MEAN at the SOA exit. These systematic signal asymmetries originating from steeper and more gradual pulse edges are the root cause of the deterministic timing jitter depending on input data characteristics. When pulses of unequal peak power arrive at the input of the SOA, each pulse is displaced by a different amount resulting in a mean arrival time deviation at the output. Thus, when intensitymodulated pulses are injected into the SOA, and assuming the pulse period is greater than the SOA-gain recovery time, the different peak power levels will generate different dips in the SOA gain that cause different pulse shifts, leading to varying timing jitter values.

Apart from the peak position deviation of the optical pulses, SOAs can also induce pulse peak power equalization of incoming intensity-modulated optical pulses. This can, in turn, yield in a reduction in intensity modulation of the optical pulses at the SOA output. Assuming, again, that the SOA gain recovery time is faster than the pulse period, the pulse peak power equali‐ zation originates from the amplification dissimilarities arising between the low and high pulse peak powers. A high gain is received by the lower peak-powered pulses whereas a lower gain is experienced by the higher peak-powered pulses, resulting in nearly power-equivalent amplified pulses obtained at the SOA exit. As a result, the modulation depth index of the input pulses will always be higher than the respective outputs of the amplifier pointing out an intensity modulation reduction of the exiting optical pulse stream.

The following analysis aims to provide a theoretical insight into the origin of the deterministic timing jitter and elucidate the pulse peak power equalization mechanism during the SOA amplification of intensity-modulated optical pulses. Considering the amplifier as a spatially concentrated device, the instantaneous amplifier gain *G*(*t*) experienced by each pulse entering the SOA is expressed as [13, 20]:

$$G\left(t\right) = 1 / \left[1 - \left(1 - 1 / G\_0\right) \cdot \exp\left(-tI\_{\ln}\left(t\right) / tI\_{\text{sat}}\right)\right] \tag{1}$$

where *G*0 represents the SOA steady-state gain and *Uin*(*t*) the accumulated injected pulse energy given by:

$$\mathrm{CLI}\_{\mathrm{in}}\left(t\right) = \bigwedge\_{\cdots \neq \cdots} \mathrm{P}\_{\mathrm{in}}\left(t\right) \cdot dt \tag{2}$$

and the *U*sat is the well-known saturation energy of the device. To this end, the output pulse exiting the SOA can be calculated as:

$$P\_{\rm out} \begin{pmatrix} t \\ \end{pmatrix} = P\_{\rm in} \begin{pmatrix} t \\ \end{pmatrix} \cdot G \begin{pmatrix} t \\ \end{pmatrix} \tag{3}$$

by defining *P*in(*t*) as the input pulse power*.*

#### **2.1. Deterministic timing jitter analysis**

Considering Gaussian pulses as input to the SOA, the input pulse power is defined as *<sup>P</sup>*in(*t*)=*Pp* <sup>⋅</sup> exp( <sup>−</sup>*<sup>t</sup>* <sup>2</sup> / *<sup>T</sup>*<sup>0</sup> 2 ) with peak power denoted as *P*p and 1/*e* pulsewidth equal to *T*0. The mean arrival time *T*MEAN for every individual pulse is calculated [21, 22] as:

$$T\_{\rm MEAN} = \frac{\int\_{-\infty}^{+\infty} t \cdot P\_{\rm out} \left(t\right) \cdot dt}{\iint\_{\rm total}} = \frac{\int\_{-\infty}^{+\infty} t \cdot P\_{\rm in} \left(t\right) \cdot G \left(t\right) \cdot dt}{\int\_{-\infty}^{+\infty} P\_{\rm in} \left(t\right) \cdot G \left(t\right) \cdot dt} = \frac{A}{B} \tag{4}$$

where *U*total is the total output pulse energy. By replacing *P*in(*t*) and *G*(*t*) with their respective expressions, we obtain *A* and *B* in Eq. (5) and Eq. (6), respectively.

Semiconductor Optical Amplifier (SOA)–based Amplification of Intensity-modulated Optical Pulses... http://dx.doi.org/10.5772/61712 113

$$A = \overset{\ast}{\int}\_{-\boldsymbol{\sigma}}^{\boldsymbol{\sigma}} \boldsymbol{P}\_{\text{p}} \cdot \exp\left(-\frac{t^{2}}{T\_{0}^{2}}\right) \cdot \left[1 - \left(1 - \frac{1}{G\_{0}}\right) \cdot \exp\left[\frac{-\boldsymbol{P}\_{\text{p}} \cdot \int\_{-\boldsymbol{\sigma}}^{t} \exp\left(-\frac{t^{2}}{T\_{0}^{2}}\right) \cdot dt}{\boldsymbol{U}\_{\text{sat}}}\right]\right]^{-1} \cdot dt \tag{5}$$

peak powers. A high gain is received by the lower peak-powered pulses whereas a lower gain is experienced by the higher peak-powered pulses, resulting in nearly power-equivalent amplified pulses obtained at the SOA exit. As a result, the modulation depth index of the input pulses will always be higher than the respective outputs of the amplifier pointing out an

The following analysis aims to provide a theoretical insight into the origin of the deterministic timing jitter and elucidate the pulse peak power equalization mechanism during the SOA amplification of intensity-modulated optical pulses. Considering the amplifier as a spatially concentrated device, the instantaneous amplifier gain *G*(*t*) experienced by each pulse entering

( ) ( ) ( ( ) ) = -- × - é ù

( ) ( ) -¥

= × in ò in *t*

where *G*0 represents the SOA steady-state gain and *Uin*(*t*) the accumulated injected pulse

and the *U*sat is the well-known saturation energy of the device. To this end, the output pulse

Considering Gaussian pulses as input to the SOA, the input pulse power is defined as

( ) ( ) ( )

¥

*U B P t G t dt*

¥

where *U*total is the total output pulse energy. By replacing *P*in(*t*) and *G*(*t*) with their respective

 ¥

 ¥

ò

× × × ×× == =

out in

total in *t P t dt t P t G t dt <sup>A</sup> <sup>T</sup>*

+ -

mean arrival time *T*MEAN for every individual pulse is calculated [21, 22] as:

+ + - -

ò ò

expressions, we obtain *A* and *B* in Eq. (5) and Eq. (6), respectively.

ë û <sup>0</sup> in sat *G t* 1 / 1 1 1 / exp / *G UtU* (1)

*U t P t dt* (2)

*P t P t Gt* out in ( ) = × ( ) ( ) (3)

) with peak power denoted as *P*p and 1/*e* pulsewidth equal to *T*0. The

( ) ( )

× ×

(4)

intensity modulation reduction of the exiting optical pulse stream.

the SOA is expressed as [13, 20]:

112 Some Advanced Functionalities of Optical Amplifiers

exiting the SOA can be calculated as:

by defining *P*in(*t*) as the input pulse power*.*

2

MEAN

¥

¥

**2.1. Deterministic timing jitter analysis**

*<sup>P</sup>*in(*t*)=*Pp* <sup>⋅</sup> exp( <sup>−</sup>*<sup>t</sup>* <sup>2</sup> / *<sup>T</sup>*<sup>0</sup>

energy given by:

$$B = \int\_{-\nu}^{\ast \nu} P\_p \cdot \exp\left(-\frac{t^2}{T\_0^2}\right) \cdot \left[1 - \left(1 - \frac{1}{G\_0}\right) \cdot \exp\left[\frac{-P\_p \cdot \int\_{-\nu}^{\prime} \exp\left(-\frac{t^2}{T\_0^2}\right) \cdot dt}{\mathcal{U}\_{\text{sat}}}\right]\right]^{-1} \cdot dt \tag{6}$$

By expanding *G*(*t*) both in the numerator *A* and in the denominator *B* in a first-order Taylor series around the center position of the pulse at *t* =0 , after some algebra, Eq. (4) becomes:

$$T\_{\rm{MEAN}} = -\frac{T\_0^2 \cdot \left(1 - \frac{1}{G\_0}\right) \cdot \left[\frac{P\_\text{p}}{\mathcal{U}\_{\rm{sat}}} \cdot \exp\left(-\frac{P\_\text{p}}{\mathcal{U}\_{\rm{sat}}} \cdot \frac{T\_0 \cdot \sqrt{\pi}}{2}\right)\right]}{2 \cdot \left[1 - \left(1 - \frac{1}{G\_0}\right) \cdot \exp\left(-\frac{P\_\text{p}}{\mathcal{U}\_{\rm{sat}}} \cdot \frac{T\_0 \cdot \sqrt{\pi}}{2}\right)\right]}\tag{7}$$

Eq. (7) determines the mean arrival time *T*MEAN of every pulse as a function of its peak power, its time duration and of the SOA steady-state gain. In Figure 1, the *T*MEAN quantity is illustrated versus a pulse peak power range of 0 mW to 20 mW, for three different SOA gain *G*0 values and a pulsewidth of 20 ps. As the center of the pulse is assumed to be at *t* =0, the time shift induced by SOA amplification process will always be leftward. As a consequence, *T*MEAN will always be a negative quantity that will continuously decrease until reaching a saturation plateau [15].

In addition, according to Figure 1, the steepness of the slope of the *T*MEAN curve indicated two distinct areas for the *T*MEAN based on the peak power: area A, corresponding to the nonsatu‐ rated SOA gain regime, where the SOA still responds radically to input peak power resulting in enhanced timing jitter values for the amplified pulses, and area B, where the SOA operates in its strongly saturated gain region. In the last case, the curve of *T*MEAN decreases smoothly, mitigating in this way the differences of mean arrival time *T*MEAN compared to SOA operation in area A and leading to lower timing jitter values [15].

The monotonic slope of mean arrival time *T*MEAN implies that the lowest and highest values are obtained for the corresponding lowest and highest peak power input pulses [15]. In case the *P*p values are within a finite set between minimum and maximum values, the peak-to-peak

**Figure 1.** Theoretical mean arrival time (*T*MEAN ) vs. pulse peak power values. Insets of the figure show the eyedia‐ grams of a jitter-free intensity-modulated signal before entering the SOA amplifier and the output signals when the SOA operates in the nonsaturation (area A) and in the strongly saturated regime (area B), respectively.

deterministic timing jitter is determined as the difference between the respective minimum and maximum mean arrival time values *T*MEAN [15], as shown in Eq. (8):

$$J\_{\rm pp}^D = T\_{\rm MEAN}^{\rm max} - T\_{\rm MEAN}^{\rm min} \tag{8}$$

where *T*MEAN min <sup>=</sup>*T*MEAN(*P*<sup>p</sup> min) and *T*MEAN max <sup>=</sup>*T*MEAN(*P*<sup>p</sup> max).

An interesting conclusion for deterministic timing jitter can be drawn by expressing the peak power *P*p in logarithmic instead of linear scale. Figure 2 is a graphical representation of Eq. (7), showing the dependence of *T*MEAN on peak power values, when the latter is expressed in dBm. As illustrated, the curved sections for peak power values lower than –5 dBm, and higher than 0 dBm can be very well approximated by second-degree polynomials.

**Figure 2.** a) Mean arrival time ( *T*MEAN) vs. pulse peak power expressed in dBm for Go = 28 dB and 30-ps-long pulses. Dashed lines denote the fit of *T*MEAN with second-degree polynomials. Linearity of deterministic timing jitter denot‐ ed by b factor of Eq. (12) versus (b) 1/e pulsewidth and (c) SOA gain levels, showing the dependence of *T*MEAN on peak power values, the latter expressed in dBm.

Thus, Eq. (7) can be expanded into a second-order Taylor series around a reference peak power *P*REF , again expressed in dBm. Moreover, by substituting in Eq. (8), the values of *T*MEAN corresponding to the highest and lowest peak power pulses *P*<sup>p</sup> max(dBm) and *P*<sup>p</sup> min(dBm), respectively, the deterministic timing jitter can now be written as shown in Eq. (9):

$$\boldsymbol{\dot{\rho}}\_{P\_{\text{p}}}^{D} = \frac{1}{2!} \cdot \boldsymbol{a} \cdot \left[ \left( P\_{\text{p}}^{\text{max}} \left( \text{dBm} \right) - P\_{\text{REF}} \left( \text{dBm} \right) \right)^{2} - \left( P\_{\text{p}}^{\text{min}} \left( \text{dBm} \right) - P\_{\text{REF}} \left( \text{dBm} \right) \right)^{2} \right] + \boldsymbol{b} \cdot \left( P\_{\text{p}}^{\text{max}} \left( \text{dBm} \right) - P\_{\text{P}}^{\text{min}} \left( \text{dBm} \right) \right) \tag{9}$$

where *a* = *<sup>d</sup>* <sup>2</sup>*<sup>J</sup> <sup>p</sup> p D <sup>d</sup> <sup>P</sup>* <sup>2</sup> <sup>|</sup> *P*=*PREF* (*dBm*) and b = *d J <sup>p</sup> p D dP* <sup>|</sup> *<sup>P</sup>*=*PREF* (*dBm*) Eq. (9) can be further simplified using some straightforward algebra into:

deterministic timing jitter is determined as the difference between the respective minimum

**Figure 1.** Theoretical mean arrival time (*T*MEAN ) vs. pulse peak power values. Insets of the figure show the eyedia‐ grams of a jitter-free intensity-modulated signal before entering the SOA amplifier and the output signals when the

max).

*JT T* (8)

= - max min pp MEAN MEAN

max <sup>=</sup>*T*MEAN(*P*<sup>p</sup>

An interesting conclusion for deterministic timing jitter can be drawn by expressing the peak power *P*p in logarithmic instead of linear scale. Figure 2 is a graphical representation of Eq. (7), showing the dependence of *T*MEAN on peak power values, when the latter is expressed in dBm. As illustrated, the curved sections for peak power values lower than –5 dBm, and higher

**Figure 2.** a) Mean arrival time ( *T*MEAN) vs. pulse peak power expressed in dBm for Go = 28 dB and 30-ps-long pulses. Dashed lines denote the fit of *T*MEAN with second-degree polynomials. Linearity of deterministic timing jitter denot‐ ed by b factor of Eq. (12) versus (b) 1/e pulsewidth and (c) SOA gain levels, showing the dependence of *T*MEAN on

and maximum mean arrival time values *T*MEAN [15], as shown in Eq. (8):

SOA operates in the nonsaturation (area A) and in the strongly saturated regime (area B), respectively.

*D*

than 0 dBm can be very well approximated by second-degree polynomials.

min) and *T*MEAN

where *T*MEAN

min <sup>=</sup>*T*MEAN(*P*<sup>p</sup>

114 Some Advanced Functionalities of Optical Amplifiers

peak power values, the latter expressed in dBm.

$$\mathbf{J}\_{p\_{\mathrm{p}}}^{D} = \frac{1}{2!} \cdot a \cdot \Delta P \text{(dB)} \cdot \left[ P\_{\mathrm{p}}^{\mathrm{max}} \left( \mathrm{dBm} \right) + P\_{\mathrm{p}}^{\mathrm{min}} \left( \mathrm{dBm} \right) - 2 \cdot P\_{\mathrm{REF}} \left( \mathrm{dBm} \right) \right] + b \cdot \Delta P \text{(dB)} \tag{10}$$

where Δ*P* =*P*<sup>p</sup> max <sup>−</sup>*P*<sup>p</sup> min is the intensity modulation expressed in dB. By selecting the *P*REF(dBm) value to be the mid-point between the minimum *P*<sup>p</sup> min and maximum *P*<sup>p</sup> max peak power level, so that *P*<sup>p</sup> max(dBm)=*P*REF(dBm) <sup>+</sup> <sup>Δ</sup>*P*(*dB*)/ 2. and *P*<sup>p</sup> min(dBm)=*P*REF(dBm)−Δ*P*(dB) / 2, the quantity contained in the brackets becomes zero and the deterministic timing jitter expression turns into:

$$J\_{p\_\rho}^D = \mathbf{b} \cdot \Delta P \text{(dB)}\tag{11}$$

This formula reveals a linear relationship between deterministic timing jitter and intensity modulation with the linearity factor b provided by Eq. (12).

$$\begin{pmatrix} -T\_0^2 \cdot e^{\frac{\text{hux}}{10} \cdot \text{s} \cdot \text{s}} \cdot \left( G\_0 - 1 \right) \cdot \\\\ \cdot \left( 2 \cdot G\_0 - 2 \cdot G\_0 \cdot e^{\frac{T\_0 \cdot \text{s} \cdot \text{f}\_{\text{sat}} \cdot \text{s}}{2}} + G\_0 \cdot T\_0 \cdot \sqrt{\pi} \cdot \text{U}\_{\text{sat}} \cdot e^{\frac{\text{hux} \cdot \left( \text{dħm} \cdot \text{l} \cdot \text{s} \right)}{10} + \frac{T\_0 \cdot \sqrt{\pi} \cdot \text{d} \cdot \text{s}}{2}} - 2 \right) \cdot \\\\ \cdot \left( \mathbf{40 \cdot \text{U}}\_{\text{sat}} \cdot \left( G\_0 \cdot e^{\frac{T\_0 \cdot \text{f}\_{\text{sat}} \cdot \text{s}}{2}} - G\_0 + 1 \right)^2 \right)^{-1} \end{pmatrix} \tag{12}$$

By plotting Eq. (12) for different pulsewidths and SOA gain levels as shown in Figure 2(b) and in Figure 2(c), respectively, the absolute value of *b* increases with *T*0 and *G*<sup>0</sup> for a given *P*REF (dBm) value. This indicates that higher jitter values are obtained for higher pulsewidths and higher SOA gains when the same intensity modulation level and the same *P*REF (dBm) values are used.

#### **2.2. Pulse peak power equalization analysis**

By defining *T* as the bit period, *Pp* as the average peak power value across the whole control signal sequence, Ω as the modulation frequency and m as the modulation depth index, the peak power of each *k*-th individual pulse of an intensity-modulated clock pulse sequence entering the SOA is given by *P*<sup>p</sup> *<sup>k</sup>* <sup>=</sup> *<sup>P</sup>*<sup>p</sup> <sup>⋅</sup> <sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>⋅</sup> cos(Ω⋅ *<sup>k</sup>* <sup>⋅</sup>*<sup>T</sup>* ) . Since multilevel clock puls are considered to enter the SOA as input, the modulation depth index can be determined by their discrete levels. By substituting the intensity-modulated clock pulse sequence *P*<sup>p</sup> *k* in (2), *U*in(*t*) is transformed as follows:

$$\mathrm{d}I\_{\mathrm{in}}\left(t\right) = \int\_{-\mathrm{e}}^{t} P\_{\mathrm{in}}\left(t\right) \cdot dt = P\_{\mathrm{p}} \cdot \left(1 + m \cdot \cos\left(\Omega \cdot k \cdot T\right)\right) \cdot \int\_{-\mathrm{e}}^{t} a\left(t'\right) \cdot dt'\tag{13}$$

where *a*(*t* ') denotes the pulse waveform. Eq. (1) shows that the gain saturates to a minimum value until the whole pulse energy has passed through the amplifier. However, it is assumed that the gain recovers back to its steady-state value before the next pulse enters the amplifier. As a result, Eq. (1) is valid for the whole bit sequence, allowing in this way for the replacement of the time-dependent integral *∫ <sup>a</sup>*(*<sup>t</sup>* ') <sup>⋅</sup>*dt* ' contained in Eq. (13) with a time-independent constant value *A* that corresponds to the total area contained in the pulse waveform [20]. To this end, the output intensity-modulated clock pulse sequence *Po*/ *<sup>p</sup>* after using Eq. (1) and Eq. (13) can be expressed as follows:

$$P\_{o/p}\left(m\right) = \frac{P\_p \cdot \left(1 + m \cdot \cos\left(\Omega \cdot k \cdot T\right)\right)}{\left[1 - \left(1 - \left(1 \wedge \mathbb{G}\_0\right)\right) \cdot \exp\left(-P\_p \cdot \left(1 + m \cdot \cos\left(\Omega \cdot k \cdot T\right)\right) \cdot A \cdot \left(U\_{\text{sat}}\right)\right)\right]}\tag{14}$$

Eq. (14) depicts that *Po*/ *<sup>p</sup>* is a function of the SOA gain *G*0, the saturation energy *U*sat and the pulse peak power *P*<sup>0</sup> ⋅ (1 + *m*⋅ cos(Ω⋅ *k* ⋅*T* )). By expanding (14) in a first-order Taylor series around *m*=0 the output pulse peak power can be written as the sum of a *dc* signal component and an oscillation term *ac* at Ω :

$$P\_{p/o} \left( m \right) = P\_{o/p} \Big|\_{m=0} + \left. \frac{\partial P\_{o/p}}{\partial m} \right|\_{m=0} \cdot m \tag{15}$$

where the *dc* component is expressed as shown in (16):

$$P\_{o/p}\Big|\_{u=0} = \frac{P\_p}{1 - \left(1 - \frac{1}{G\_0}\right) \cdot \exp\left(-\frac{P\_p \cdot A}{\mathcal{U}\_{\text{sat}}}\right)}\tag{16}$$

and the *ac* component as

By plotting Eq. (12) for different pulsewidths and SOA gain levels as shown in Figure 2(b) and in Figure 2(c), respectively, the absolute value of *b* increases with *T*0 and *G*<sup>0</sup> for a given *P*REF (dBm) value. This indicates that higher jitter values are obtained for higher pulsewidths and higher SOA gains when the same intensity modulation level and the same *P*REF (dBm)

By defining *T* as the bit period, *Pp* as the average peak power value across the whole control signal sequence, Ω as the modulation frequency and m as the modulation depth index, the peak power of each *k*-th individual pulse of an intensity-modulated clock pulse sequence

considered to enter the SOA as input, the modulation depth index can be determined by their

discrete levels. By substituting the intensity-modulated clock pulse sequence *P*<sup>p</sup>

( ) ( ) ( ( )) ( ) - -

= × = × + × W× × × × in ò ò in <sup>p</sup> 1 cos ' ' *t t*

where *a*(*t* ') denotes the pulse waveform. Eq. (1) shows that the gain saturates to a minimum value until the whole pulse energy has passed through the amplifier. However, it is assumed that the gain recovers back to its steady-state value before the next pulse enters the amplifier. As a result, Eq. (1) is valid for the whole bit sequence, allowing in this way for the replacement

of the time-dependent integral *∫ <sup>a</sup>*(*<sup>t</sup>* ') <sup>⋅</sup>*dt* ' contained in Eq. (13) with a time-independent constant value *A* that corresponds to the total area contained in the pulse waveform [20]. To this end, the output intensity-modulated clock pulse sequence *Po*/ *<sup>p</sup>* after using Eq. (1) and Eq.

> ( ( )) ( ( ( )) ) × + × W× × <sup>=</sup> é ù - - × - × + × W× × × ë û

1 cos 1 1 1 / exp 1 cos /

*P m kT*

Eq. (14) depicts that *Po*/ *<sup>p</sup>* is a function of the SOA gain *G*0, the saturation energy *U*sat and the pulse peak power *P*<sup>0</sup> ⋅ (1 + *m*⋅ cos(Ω⋅ *k* ⋅*T* )). By expanding (14) in a first-order Taylor series around *m*=0 the output pulse peak power can be written as the sum of a *dc* signal component

> ¶ =+ × ¶ /

*PmP m*

*P*

*o p*

*p*

0 sat

=

*m*

0

( ) ( ( ))

*p*

( ) <sup>=</sup>

/ / <sup>0</sup>

*po op <sup>m</sup>*

*<sup>k</sup>* <sup>=</sup> *<sup>P</sup>*<sup>p</sup> <sup>⋅</sup> <sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>⋅</sup> cos(Ω⋅ *<sup>k</sup>* <sup>⋅</sup>*<sup>T</sup>* ) . Since multilevel clock puls are

 ¥

*G P m kT A U* (14)

*<sup>m</sup>* (15)

*U t P t dt P m k T a t dt* (13)

*k*

in (2), *U*in(*t*)

values are used.

**2.2. Pulse peak power equalization analysis**

¥

entering the SOA is given by *P*<sup>p</sup>

116 Some Advanced Functionalities of Optical Amplifiers

(13) can be expressed as follows:

and an oscillation term *ac* at Ω :

/

*o p*

*P m*

is transformed as follows:

$$\frac{\partial P\_{o/p}}{\partial m}\bigg|\_{m=0} = \left[\frac{P\_p}{1 - \left(1 - \frac{1}{G\_0}\right) \cdot \exp\left(-\frac{P\_p \cdot A}{\cdot \mathcal{U}\_{sat}}\right)} - \frac{P\_p \cdot \left(1 - \frac{1}{G\_0}\right) \cdot \frac{P\_p \cdot A}{\cdot \mathcal{U}\_{sat}} \cdot \exp\left(-\frac{P\_p \cdot A}{\cdot \mathcal{U}\_{sat}}\right)}{\left[1 - \left(1 - \frac{1}{G\_0}\right) \cdot \exp\left(-\frac{P\_p \cdot A}{\cdot \mathcal{U}\_{sat}}\right)\right]^2}\right] \cdot \cos\left(\Omega \cdot k \cdot T\right) \tag{17}$$

Dividing all terms of Eq. (15) by the *dc* component, we can calculate the modulation depth index at the output by dividing Eq. (17) by Eq. (16) and then multiplying by *m*, as shown in (18):

$$m\_{o/p} = \left\lceil ac \left( \text{component} \right) / dc \left( \text{component} \right) \right\rceil \cdot m \tag{18}$$

Finally, the modulation depth index of the output pulse peak power *mo*/ *<sup>p</sup>* is found to be:

$$m\_{o/p} = \frac{1 - \left(1 - \frac{1}{G\_0}\right) \cdot \exp\left(-\frac{P\_p \cdot A}{\mathcal{U}\_{\text{sat}}}\right) \cdot \left(1 + \frac{P\_p \cdot A}{\mathcal{U}\_{\text{sat}}}\right)}{1 - \left(1 - \frac{1}{G\_0}\right) \cdot \exp\left(-\frac{P\_{p\cdot A}}{\mathcal{U}\_{\text{sat}}}\right)} \cdot m \tag{19}$$

Eq. (19) provides a complete description of the SOA amplifier response to an injected intensitymodulated clock pulse sequence. It shows that the intensity modulation at the output is linearly related to the intensity modulation at the input and that the constant of proportionality depends on the SOA steady-state gain *G*0, the average peak power *P*<sup>p</sup> and the saturation energy *U*sat. In addition, the intensity modulation always decreases at the output of the amplifier. To relate the input and output intensity modulation indices *m* and *mo*/ *<sup>p</sup>* respectively, we define the Intensity Modulation Reduction (IMR) index as

$$\text{IMR} = 10 \cdot \log \left| m\_{o/p} / m \right| \tag{20}$$

Given that the intensity modulation depth indices *m* and *m*o/p of the input and the amplified output pulses are in principle the amplitudes of a slow varying frequency component at Ω inducing the power fluctuations on the pulses [20], Eq. (20) indicates the Intensity Modulation Reduction (IMR) of this frequency component after the SOA amplification process. In specific, the *m*o/p power level at the SOA's exit will be reduced from its respective power level at the input *m* by an amount equal to the IMR value.

**Figure 3.** Theoretical Intensity Modulation Reduction (IMR) expressed in dB vs. *U*in/*U*out values. IMR is illustrated for different SOA gain levels in the case of 30-ps-long pulses.

Figure 3 depicts the graphical representation of Eq. (20) for different SOA gains versus *U*in / *U*sat quantity. The values of IMR represent a negative quantity since the output modulation depth is always smaller than the respective values at the input. A rapid drop of IMR leading to enhanced intensity modulation suppression is illustrated in Figure 3, when *U*in / *U*sat is almost 0.04 and the pulsewidth is equal to 30 ps for all SOA gain levels. Figure 3 shows that when SOA operates in low gain level corresponding to a gain value of 20 dB, a nearly constant IMR of 8 dB is obtained. However, for higher gains, the IMR increases to reach 14 dB for a SOA gain of 31 dB. In addition, the maximum intensity modulation reduction for a SOA gain value equal to 31 dB is achieved when *U*in / *U*sat quantity takes values between 0.01 and 0.05, implying that the SOA is capable of suppressing a large power variation at its input. As *U*in increases, the IMR curves present a flat form and it is clamped to a constant level irrespective of the inserted pulse energy, after this specific *U*in threshold.

## **3. Experiment and results**

The scope of this section is to provide experimental verification of the theoretical analysis for the deterministic timing jitter and the intensity modulation reduction induced by the SOA amplification process. Figure 4 demonstrates the experimental setup that was used for measurements with different pulsewidths and SOA gain levels. It consists of a 1549.2 nm modelocked laser (TMLL) and a Ti:LiNbO-3 electro-optic modulator (MOD) driven by a 10 Gb/s pattern of alternating "1"s and "0"s, to create clock pulses at 5 GHz, so as to ensure a pulse period greater than the SOA gain recovery time (160 ps 1/e).

Given that the intensity modulation depth indices *m* and *m*o/p of the input and the amplified output pulses are in principle the amplitudes of a slow varying frequency component at Ω inducing the power fluctuations on the pulses [20], Eq. (20) indicates the Intensity Modulation Reduction (IMR) of this frequency component after the SOA amplification process. In specific, the *m*o/p power level at the SOA's exit will be reduced from its respective power level at the

**Figure 3.** Theoretical Intensity Modulation Reduction (IMR) expressed in dB vs. *U*in/*U*out values. IMR is illustrated for

Figure 3 depicts the graphical representation of Eq. (20) for different SOA gains versus *U*in / *U*sat quantity. The values of IMR represent a negative quantity since the output modulation depth is always smaller than the respective values at the input. A rapid drop of IMR leading to enhanced intensity modulation suppression is illustrated in Figure 3, when *U*in / *U*sat is almost 0.04 and the pulsewidth is equal to 30 ps for all SOA gain levels. Figure 3 shows that when SOA operates in low gain level corresponding to a gain value of 20 dB, a nearly constant IMR of 8 dB is obtained. However, for higher gains, the IMR increases to reach 14 dB for a SOA gain of 31 dB. In addition, the maximum intensity modulation reduction for a SOA gain value equal to 31 dB is achieved when *U*in / *U*sat quantity takes values between 0.01 and 0.05, implying that the SOA is capable of suppressing a large power variation at its input. As *U*in increases, the IMR curves present a flat form and it is clamped to a constant level irrespective of the inserted

The scope of this section is to provide experimental verification of the theoretical analysis for the deterministic timing jitter and the intensity modulation reduction induced by the SOA

input *m* by an amount equal to the IMR value.

118 Some Advanced Functionalities of Optical Amplifiers

different SOA gain levels in the case of 30-ps-long pulses.

pulse energy, after this specific *U*in threshold.

**3. Experiment and results**

**Figure 4.** Experimental setup used for the deterministic timing jitter and intensity modulation reduction measure‐ ments.

In order to create an intensity-modulated pulse sequence, the clock signal is then injected into a second modulator driven by a 625 MHz sinusoidal signal that creates pulses with 8 different pulse peak power levels. The intensity-modulated clock signal is amplified via an erbium doped fiber amplifier (EDFA) in order to compensate the losses and properly adjust the required power levels of optical pulses before their introduction into the SOA. Two fiber spools of 800 m and 1225 m were employed to enable pulsewidth adjustment at 20 ps and 30 ps by exploiting the fiber dispersion. An additional CW beam at 1558.2 nm was utilized to adjust SOA gain level and as such to determine its operational regime. After setting the SOA gain to the desired value, the output pulse train was captured on a real-time oscilloscope with 16 GHz bandwidth and a jitter measurement floor of 300 fs, where the jittery pulses were collected for offline postprocessing. The experimental setup of Figure 4 was also used in order to experi‐ mentally verify the IMR graphs shown in Figure 3. By varying the CW signal inserted into the SOA in order to cover a broad operational SOA gain regime, the intensity modulation of the input signal, defined as the highest to the lowest pulse peak power ratio, was measured at the output of the SOA. The operation of the amplifier both in the nonsaturated regime and in the saturated regime was also ensured by properly adjusting the input signal power. By calculating the difference between the initial and the output modulation depth values, the experimental data of IMR for every different SOA gain level was obtained. The control and input signals were adjusted in terms of power and polarization by means of variable optical attenuators (VOA) and polarization controllers (PC). The SOA module was a 1.5-mm-long multiquantum well structure with a small signal gain of 31 dB. The device was driven at 450 mA and the *U*sat parameter of the SOA was found to be approximately 7 fJ. The jittery pulses that were captured on the real-time oscilloscope at 100 GSa/s for offline postprocessing. The collected optical pulses were reconstructed with a sample time resolution of Δ*t* =1.25 ps, after 8-fold upsam‐ pling. For each run, the total output timing jitter, referred to as *J* TOTAL , was calculated over 8192 pulses by means of Eq. (4) [15]. The total output timing jitter referred as *J* TOTAL consists of noise-induced random jitter and deterministic jitter [23]. The deterministic timing jitter stems from pulse edge variations depending on the input data characteristics. On that account, it is crucial to separate the stochastic contribution of random jitter from the deterministic process that is only responsible for timing variations proportional to the pulses' intensity modulation. Thus, deterministic jitter is calculated as a peak-to-peak value (*J*pp *<sup>D</sup>*) between a minimum and a maximum value since a probabilistic distribution cannot be applied. On the contrary, random jitter is determined as the root-mean-square (rms) value of a normal distribution (*J*pp *<sup>R</sup>* ). By using well-known equations of converting the root-mean-square (rms) to peak-to-peak values [24, 25] the total timing jitter *J* TOTAL can be finally calculated as the following:

$$J^{\text{TOTAL}} = J\_{\text{pp}}^{\mathbb{R}} + J\_{\text{pp}}^{D} \tag{21}$$

The total timing jitter at the output of the SOA in the absence of pulse peak power variations is uncorrelated to the timing jitter induced from an intensity-modulated pulse sequence [15]. As such, it represents the accumulated random timing jitter of our experimental system:

$$J\_{\rm pp}^R = J^{\rm TOTAL} \Big|\_{\rm intensity \, modulation = 0 \, dB} \tag{22}$$

Based on this assumption, the deterministic timing jitter induced by the SOA amplification process can be calculated by subtracting the random jitter measurement floor from *J* TOTAL when the input pulse sequence has a given intensity modulation [15]. Table 1 summarizes the timing jitter values for power-equalized pulses when the pulsewidth values are 20 ps and 30 ps and the SOA gains equal to 20 dB and 31 dB, respectively.


**Table 1.** Timing jitter values for power-equalized pulses at the input and at the output of the SOA

#### **3.1. Deterministic timing jitter: Theoretical and experimental results**

8192 pulses by means of Eq. (4) [15]. The total output timing jitter referred as *J* TOTAL consists of noise-induced random jitter and deterministic jitter [23]. The deterministic timing jitter stems from pulse edge variations depending on the input data characteristics. On that account, it is crucial to separate the stochastic contribution of random jitter from the deterministic process that is only responsible for timing variations proportional to the pulses' intensity

minimum and a maximum value since a probabilistic distribution cannot be applied. On the contrary, random jitter is determined as the root-mean-square (rms) value of a normal

to peak-to-peak values [24, 25] the total timing jitter *J* TOTAL can be finally calculated as the

pp pp

The total timing jitter at the output of the SOA in the absence of pulse peak power variations is uncorrelated to the timing jitter induced from an intensity-modulated pulse sequence [15]. As such, it represents the accumulated random timing jitter of our experimental system:

Based on this assumption, the deterministic timing jitter induced by the SOA amplification process can be calculated by subtracting the random jitter measurement floor from *J* TOTAL when the input pulse sequence has a given intensity modulation [15]. Table 1 summarizes the timing jitter values for power-equalized pulses when the pulsewidth values are 20 ps and 30

**SOA Input SOA Output**

20

31

0.634 5.065 5.180

4.460 6.596 5.779

**Pulsewidth (ps) Random Jitter (ps) Gain (dB) Random Jitter Floor** *J***PP**

*<sup>R</sup>* ). By using well-known equations of converting the root-mean-square (rms)

*R D J JJ* (21)

*J J* (22)

*<sup>D</sup>*) between a

*<sup>R</sup>* **(ps)**

4.356 4.391

6.082 4.223

modulation. Thus, deterministic jitter is calculated as a peak-to-peak value (*J*pp

= + TOTAL

<sup>=</sup> <sup>=</sup> TOTAL pp intensity modulation 0 dB

*R*

ps and the SOA gains equal to 20 dB and 31 dB, respectively.

0.585

4.048

**Table 1.** Timing jitter values for power-equalized pulses at the input and at the output of the SOA

distribution (*J*pp

120 Some Advanced Functionalities of Optical Amplifiers

Nonsaturation

20

20

<sup>30</sup> *<sup>J</sup>*rms

<sup>30</sup> *<sup>J</sup>*PP

Saturati on

*R*

*R*

following:

Figure 5(a) and 5(b) show the eyediagrams of a 10 dB intensity-modulated input signal with 20 ps and 30 ps pulsewidths, respectively. Figure 5(c) and 5(d) depict the output eyediagrams for the two pulsewidths when the amplifier operates in the nonsaturated regime. Figure 5(e) and 5(f) illustrate similar results for the two pulsewidths in case the SOA is operated in the strongly saturated gain region. The experimental average peak power values for the eyedia‐ grams obtained in Figure 5 are shown in Table 2. In the eyediagrams of Figure 5(c–f), the deterministic jitter is masked under the contribution of total jitter including the accumulated random jitter of the system as well. The irregular shapes of the output eye diagrams reveal, however, the pulse shape distortion that triggers the deterministic timing jitter.


**Table 2.** Experimental values for the obtained SOA eyediagrams in Figure 5.

Figure 5(g–j) depicts the experimental and theoretical results of the deterministic timing jitter versus input signal intensity modulation expressed in dB, for different gain levels, saturation regimes of the SOA and pulsewidths.

**Figure 5.** (a), (b) Eyediagram of an input optical signal with 10 dB intensity modulation with a pulsewidth of (a) 20 ps and (b) 30 ps, and the respective SOA output when the amplifier operates in the (c), (d) nonsaturated and (e),(f) strong‐ ly saturated regimes. The SOA gain level is equal to 20 dB for (a–f). Timescale for (a–f): 10 ps/div. Experimental and theoretical results for the deterministic timing jitter vs. pulse peak power modulation (intensity modulation) expressed in dB, for 30 ps pulsewidth and 31 dB SOA gain in (g) nonsaturated and (h) strongly saturated regimes, and for 20 dB SOA gain and nonsaturated region with (i) 30 ps and (j) 20 ps pulsewidths, respectively.

According to Eq. (11), the theoretical deterministic timing jitter depends linearly on intensity modulation levels. On that account, a linear fit was applied to the experimental data revealing good agreement between theoretical and experimental results obtained in all cases [15]. Figure 5(g) and (h) show theoretical deterministic timing jitter results obtained by applying Eq. (7) into Eq. (8), as well as the experimental data with their linear fit for 30 ps pulsewidth and 31 dB SOA gain level. The average pulse peak power values used in this case were 11 μW and 235 μW for unsaturated and saturated SOA operations, respectively. The graphs reveal a reduction of deterministic timing jitter in excess of 25% for the case of the SOA saturated operational regime. Figure 5(i) illustrates deterministic timing jitter evolution versus intensity modulation levels for 20 dB SOA gain using an average pulse peak power value of 34 μW. When compared with Figure 5(g), a decrease of the deterministic timing jitter values with the SOA gain level is evident. Finally, Figure 5(i) and 5(j) depict the deterministic timing jitter results for 30 ps and 20 ps pulsewidths, respectively, when all other operating parameters are the same, confirming that shorter pulses generate lower deterministic timing jitter levels [30 W average pulse peak power values for Fig. 5(j)]. In all cases, good qualitative and quantitative agreement between experiment and theory was achieved retaining the same deterministic timing jitter trend.

#### **3.2. Intensity modulation reduction: Theoretical and experimental results**

Figure 6 depicts the theoretical and experimental results for the output *m*o/p versus the input *m* modulation depth indices and the *m*o/p / *m* versus the input pulse energy *U*in of optical pulses inserted in the SOA. The theoretical curves are denoted by solid lines, and the experi‐ mental observations by bullets. A linear fit was also applied to the experimental data in Figure 6(a), 6(b) and 6(d), due to the linear nature of the IMR indicated by Eq. (19) and it is represented by a dashed line. As can be noticed in Figure 6(a) and 6(b), the output modulation depth index depends linearly on the input modulation depth index when the SOA gain levels equal to 20 dB and 28 dB, respectively. In both cases, the amplifier operates in the nonsaturated region with the *U*in values reaching up to 5 fJ and 0.5 fJ, respectively. Figure 6(c) demonstrates the *m*o/p / *m* ratio versus the input pulse energy *U*in for SOA gain levels of 20 dB, 28 dB and 31 dB. As the gain level rises from 20 dB to 31 dB, the steepness of the slope increases and the curve shifts closer to the axis. In the case of gain equal to 31 dB, both unsaturated and saturated SOA experimental observations are shown in Figure 6(d) revealing higher intensity modulation reduction for strongly saturated SOA with the *U*in reaching 2 fJ. Figure 6(d) presents in detail the output *m*o/p versus the input *m* modulation indices for the two operational SOA regimes for 31 dB gain level. The *U*in value is directly associated to the SOA operational regime and imposes the slope of the *m*o/p versus *m* curve resulting in low or high variation between input and output modulation depth that in turn yields low- or high-intensity modulation reduction values. It can be observed that the slope of the curve is smaller in the case of a saturated SOA corresponding to *U*in equal to 2 fJ in comparison with the unsaturated SOA referring to a *U*in value equal to 0.2 fJ.

Semiconductor Optical Amplifier (SOA)–based Amplification of Intensity-modulated Optical Pulses... http://dx.doi.org/10.5772/61712 123

**Figure 6.** (a), (b) and (d) experimental and theoretical results for output modulation depth index versus input modula‐ tion depth index of intensity-modulated optical pulses for G0=20dB, G0 =28dB and G0=31dB of the SOA. (d) Results in both nonsaturation and saturation regime of the SOA and (c) intensity modulation reduction (*m*o/p*/m*in) for the three SOA gain levels vs. *U*in. In all cases, the bullets represent experimental measurements, the solid lines the respective theoretical curves and the dashed lines the experimental fit.

Comparison between dashed and solid lines in Figure 6(a–d) shows good agreement between theory and experiment and indicates the SOA potential to provide increased pulse peak power equalization at its output, when operating the amplifier in the saturation regime.

### **4. Discussion**

According to Eq. (11), the theoretical deterministic timing jitter depends linearly on intensity modulation levels. On that account, a linear fit was applied to the experimental data revealing good agreement between theoretical and experimental results obtained in all cases [15]. Figure 5(g) and (h) show theoretical deterministic timing jitter results obtained by applying Eq. (7) into Eq. (8), as well as the experimental data with their linear fit for 30 ps pulsewidth and 31 dB SOA gain level. The average pulse peak power values used in this case were 11 μW and 235 μW for unsaturated and saturated SOA operations, respectively. The graphs reveal a reduction of deterministic timing jitter in excess of 25% for the case of the SOA saturated operational regime. Figure 5(i) illustrates deterministic timing jitter evolution versus intensity modulation levels for 20 dB SOA gain using an average pulse peak power value of 34 μW. When compared with Figure 5(g), a decrease of the deterministic timing jitter values with the SOA gain level is evident. Finally, Figure 5(i) and 5(j) depict the deterministic timing jitter results for 30 ps and 20 ps pulsewidths, respectively, when all other operating parameters are the same, confirming that shorter pulses generate lower deterministic timing jitter levels [30 W average pulse peak power values for Fig. 5(j)]. In all cases, good qualitative and quantitative agreement between experiment and theory was achieved retaining the same deterministic

**3.2. Intensity modulation reduction: Theoretical and experimental results**

Figure 6 depicts the theoretical and experimental results for the output *m*o/p versus the input *m* modulation depth indices and the *m*o/p / *m* versus the input pulse energy *U*in of optical pulses inserted in the SOA. The theoretical curves are denoted by solid lines, and the experi‐ mental observations by bullets. A linear fit was also applied to the experimental data in Figure 6(a), 6(b) and 6(d), due to the linear nature of the IMR indicated by Eq. (19) and it is represented by a dashed line. As can be noticed in Figure 6(a) and 6(b), the output modulation depth index depends linearly on the input modulation depth index when the SOA gain levels equal to 20 dB and 28 dB, respectively. In both cases, the amplifier operates in the nonsaturated region with the *U*in values reaching up to 5 fJ and 0.5 fJ, respectively. Figure 6(c) demonstrates the *m*o/p / *m* ratio versus the input pulse energy *U*in for SOA gain levels of 20 dB, 28 dB and 31 dB. As the gain level rises from 20 dB to 31 dB, the steepness of the slope increases and the curve shifts closer to the axis. In the case of gain equal to 31 dB, both unsaturated and saturated SOA experimental observations are shown in Figure 6(d) revealing higher intensity modulation reduction for strongly saturated SOA with the *U*in reaching 2 fJ. Figure 6(d) presents in detail the output *m*o/p versus the input *m* modulation indices for the two operational SOA regimes for 31 dB gain level. The *U*in value is directly associated to the SOA operational regime and imposes the slope of the *m*o/p versus *m* curve resulting in low or high variation between input and output modulation depth that in turn yields low- or high-intensity modulation reduction values. It can be observed that the slope of the curve is smaller in the case of a saturated SOA corresponding to *U*in equal to 2 fJ in comparison with the unsaturated SOA referring to a *U*in

timing jitter trend.

122 Some Advanced Functionalities of Optical Amplifiers

value equal to 0.2 fJ.

The theoretical framework and its experimental verification for both deterministic timing jitter and intensity modulation reduction analysis have relied on the assumption that every pulse experiences the same initial steady-state gain. This assumption allowed for the treatment of the pulse sequence on a per pulse basis and for the use of clock pulses for its experimental validation. However, the theoretical analysis presented here can also be extended towards calculating both these phenomena, in the case of random data patterns with intensitymodulated pulses used as the input signal in SOAs.

In the case of deterministic timing jitter, when the SOA gain recovery time is faster than the bit period, all the incoming data pulses will again experience the same steady-state gain *G*<sup>0</sup> inside the amplifier. This condition allows Eq. (7) of the mean arrival time *T*MEAN to be valid. But even if the SOA gain recovery time is slower than the bit period, Eq. (7) can be exploited for calculating the mean arrival time *T*MEAN of the data pulses and, subsequently, the deter‐ ministic timing jitter. In this case, the use of a random incoming data pattern into the SOA will actually result in different gain levels experienced by every individual optical pulse. On that account, *G*<sup>0</sup> should then be treated as an additional variable in Eq. (7) with its values residing within a certain range Δ*G*0. This actually turns relationship (7) into a two-variable function, assuming a given pulsewidth and a constant *U*sat parameter [15].

According to Figure 1, the same pulse peak power level results in a lower absolute value for the pulse mean arrival time when a lower gain value is perceived by the pulse. In Figure 1, for example, the absolute value of *T*MEAN for a gain of 20 dB is always lower than the respective value for a SOA gain of 28 dB, which in turn is always lower than the respective value for a 30 dB SOA gain. This indicates that the deterministic jitter in case of different gain levels perceived by every pulse, as will be the case with random incoming data patterns and gain recovery times slower than the bit period, will be always slightly higher compared to the deterministic jitter values induced by the same pulse sequence when the SOA gain recovery time is faster than the bit period. For example, when the incoming data pulse with the lowest peak power level enters the amplifier after a long sequence of "1"s, it experiences the lowest SOA gain among all data pulses yielding, in this way, the lowest absolute value for its mean arrival time at the SOA output. At the same time, the highest data pulse comes after a long sequence of "0"s, so that it actually experiences the full gain of the amplifier, resulting in the highest absolute value among all data pulses for its mean arrival time *T*MEAN. This scenario can certainly occur when a truly random data pattern with an intensity modulation that follows a certain statistical distribution will be injected into the SOA [15]. To this end, Eq. (7) indicates that higher deterministic timing jitter values should be expected in this case.

Following the same rationale, Eq. (19) of the modulation depth index of the output pulse peak power *m*o/p can also turn into a two-variable function (*G*0, *P*p) for a certain pulse waveform A and a constant *U*sat parameter. In this way, it enables its utilization in cases of random data pattern when the SOA gain recovery time is greater than the bit period. Again, the worst-case scenario in terms of the intensity modulation level of the output pulses will take place when the pulses with the smallest peak power level follow a long sequence of "1"s and the pulses with the highest peak power level come after a long sequence of "0"s. The smallest pulses will experience the lowest gain while the highest pulses will perceive the full gain of the amplifier resulting in a less optimal peak power equalization compared to the respective case where the SOA gain recovers, and all pulses receive the same steady-state gain *G*0. In such a traffic scenario, the random data pattern will limit the power equalization dynamics of the SOA, since the bit randomness will affect its ability to provide the same gain to every pulse and will lead to varying gain values that will be imprinted on the amplified optical pulses.

## **5. Conclusion**

Research interest in semiconductor optical amplifiers (SOAs) has been lately renewed since SOAs appear as the most preferable on-chip amplifier option in many key network subsystems. Although a concerted research effort on SOA-based devices spanning the last 20 years, has revealed most of their underlying amplification secrets, SOA effects on intensity-modulated optical pulses in terms of timing jitter and pulse peak power equalization have not yet been consolidated in a detailed analytical framework. On that account, we have presented in this chapter, a holistic theoretical framework verified by experimental results that establishes for the first time a systematic methodology for the deterministic timing jitter and peak power equalization estimation in case of intensity-modulated optical pulses entering the SOA. Experimental and theoretical results reveal a linear relationship between deterministic timing jitter and intensity modulation levels when the SOA gain recovery time is shorter than the bit period. The pulse mean arrival time is calculated as a function of the pulse peak power, the pulsewidth and the SOA steady-state gain. In addition, pulse peak power equalization analysis shows that intensity modulation at output is linearly related to the intensity modulation at the input and the constant of proportionality depends on the SOA steady-state gain *G*0, the pulse peak power *P*<sup>p</sup> and the saturation energy *U*sat. Both deterministic timing jitter and intensity modulation reduction formulas derived in the proposed theoretical analysis, enable a quali‐ tative and quantitative insight into the SOA performance when intensity-modulated optical pulses are inserted into the amplifier.

## **Acknowledgements**

for calculating the mean arrival time *T*MEAN of the data pulses and, subsequently, the deter‐ ministic timing jitter. In this case, the use of a random incoming data pattern into the SOA will actually result in different gain levels experienced by every individual optical pulse. On that account, *G*<sup>0</sup> should then be treated as an additional variable in Eq. (7) with its values residing within a certain range Δ*G*0. This actually turns relationship (7) into a two-variable function,

According to Figure 1, the same pulse peak power level results in a lower absolute value for the pulse mean arrival time when a lower gain value is perceived by the pulse. In Figure 1, for example, the absolute value of *T*MEAN for a gain of 20 dB is always lower than the respective value for a SOA gain of 28 dB, which in turn is always lower than the respective value for a 30 dB SOA gain. This indicates that the deterministic jitter in case of different gain levels perceived by every pulse, as will be the case with random incoming data patterns and gain recovery times slower than the bit period, will be always slightly higher compared to the deterministic jitter values induced by the same pulse sequence when the SOA gain recovery time is faster than the bit period. For example, when the incoming data pulse with the lowest peak power level enters the amplifier after a long sequence of "1"s, it experiences the lowest SOA gain among all data pulses yielding, in this way, the lowest absolute value for its mean arrival time at the SOA output. At the same time, the highest data pulse comes after a long sequence of "0"s, so that it actually experiences the full gain of the amplifier, resulting in the highest absolute value among all data pulses for its mean arrival time *T*MEAN. This scenario can certainly occur when a truly random data pattern with an intensity modulation that follows a certain statistical distribution will be injected into the SOA [15]. To this end, Eq. (7) indicates

assuming a given pulsewidth and a constant *U*sat parameter [15].

124 Some Advanced Functionalities of Optical Amplifiers

that higher deterministic timing jitter values should be expected in this case.

to varying gain values that will be imprinted on the amplified optical pulses.

**5. Conclusion**

Following the same rationale, Eq. (19) of the modulation depth index of the output pulse peak power *m*o/p can also turn into a two-variable function (*G*0, *P*p) for a certain pulse waveform A and a constant *U*sat parameter. In this way, it enables its utilization in cases of random data pattern when the SOA gain recovery time is greater than the bit period. Again, the worst-case scenario in terms of the intensity modulation level of the output pulses will take place when the pulses with the smallest peak power level follow a long sequence of "1"s and the pulses with the highest peak power level come after a long sequence of "0"s. The smallest pulses will experience the lowest gain while the highest pulses will perceive the full gain of the amplifier resulting in a less optimal peak power equalization compared to the respective case where the SOA gain recovers, and all pulses receive the same steady-state gain *G*0. In such a traffic scenario, the random data pattern will limit the power equalization dynamics of the SOA, since the bit randomness will affect its ability to provide the same gain to every pulse and will lead

Research interest in semiconductor optical amplifiers (SOAs) has been lately renewed since SOAs appear as the most preferable on-chip amplifier option in many key network subsystems.

This work has been supported in part by the European Commission through FP7-ICT-IP project PhoxTrot (contract no. 318240) and FP7 MC-IAPP project COMANDER (contract no. 612257).

## **Author details**

T. Alexoudi1,2\*, G.T. Kanellos2 , S. Dris3 , D. Kalavrouziotis3 , P. Bakopoulos3 , A. Miliou1,2 and N. Pleros1,2

\*Address all correspondence to: theonial@iti.gr

1 Information & Technologies Institute, Center for Research & Technology Hellas, Thessaloniki, Greece

2 Department of Informatics, Aristotle University of Thessaloniki, Thessaloniki, Greece

3 School of Electrical Engineering and Computer Engineering, National Technical University of Athens, Athens, Greece

## **References**


[12] M. Settembre et al. Cascaded optical communication systems with in-line semicon‐ ductor optical amplifiers. IEEE/OSA Journal of Lightwave Technology. 1997; 15(6): 962–967. DOI: 10.1109/50.588666

**References**

2010.5549097

126 Some Advanced Functionalities of Optical Amplifiers

[1] W. Freude et al. Linear and nonlinear semiconductor optical amplifiers. In: Proceed‐ ings of 12th International Conference on Transparent Optical Networks (ICTON 2010); 27 June–1 July; Munich, Germany. IEEE; 2010. p. 1–4. DOI: 10.1109/ICTON.

[2] Dong, X. Zhang, S. Fu, J. Xu, P. Shum and D. Huang. Ultrafast all-optical signal proc‐ essing based on single semiconductor optical amplifier. Journal of Selected Topics in

[3] J. Leuthold. All-optical wavelength conversion up to 100 Gbit/s with SOA delayedinterference configuration. OSA Trends in Optics and Photonics. 2000; 44(Optical

[4] Z. Li et al. All-optical logic gates using semiconductor optical amplifier assisted by optical filter. Electronic Letters. 2005; 41(25):1397–1399. DOI: 10.1049/el:20053385 [5] G. Berrettinni, A. Simi, A. Malacarne, A. Bogoni, and L. Potí. Ultrafast integrable and reconfigurable XNOR, AND, NOR and NOT photonic logic gate. IEEE Photonics

[6] A. Hamie, A. Sharaiha, M. Guégan, and B. Puce. All-optical logic NOR gate using two-cascaded semiconductor optical amplifiers. IEEE Photonics Technology Letters.

[7] G. T. Kanellos et al. All-optical 3R burst mode reception at 40 Gb/s using 4 integrated MZI switches. IEEE/OSA Journal of Lightwave Technology. 2007; 25(1):184–192. DOI:

[8] L. Aggazi et al. Monolithic integration of the erbium-doped amplifier with siliconon-insulator waveguides. OSA Optics Express. 2010;18(26):27703–27711. DOI:

[9] C.S. Nicholes et al. An 8x8 InP monolithic tunable optical router (MOTOR) packet forwarding chip. IEEE/OSA Journal of Lightwave Technology. 2010;28(4):641–650.

[10] V.S. Pato et al. All-optical burst mode power equalizer based on cascaded SOAs for 10-Gb/s EPONs. IEEE Photonics Technology Letters. 2008;20(24):2078–2080. DOI:

[11] D. Chiaroni et al. Optical packet ring network offering bit rate and modulation for‐ mats transparency. In: Proceedings of Optical Fiber Communication (OFC) Confer‐ ence; 21–25 March; San Diego, CA, USA. IEEE; 2010. p. 1–3. DOI: 10.1364/OFC.

Quantum Electronics. 2008; 14(3):770–778. DOI: 10.1109/JSTQE.2008.916248

Technology Letters. 2006; 18(8):917–919. DOI: 10.1109/LPT.2006.873570

2002;14(10):1439–1441. DOI: 10.1109/LPT.2002.802426

10.1109/JLT.2006.888169

10.1364/OE.18.027703

DOI: 10.1109/JLT.2009.2030145

10.1109/LPT.2008.2006629

2010.OWI3

Amplifiers and Their Applications):OWB3.

