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## **Meet the editors**

Moh. Yasin received his BSc in 1990 from Airlangga University. He went on to receive MSc and PhD degrees from Gadjah Mada University in 1999 and 2010.. His research interests include fiber optic communications and sensors. Currently, he is an associate professor at the Faculty of Science and Technology, Airlangga University, Indonesia.

Hamzah Arof received his BSc from Michigan State University, and PhD from the University of Wales. Both degrees were in electrical engineering. His current research interests include signal processing and photonics. Currently he is affiliated with the Department of Electrical Engineering, University of Malaya, Malaysia.

Sulaiman Wadi Harun received the BE degree in Electrical and Electronics System Engineering from Nagaoka University of Technology, Japan in 1996. He received his MSc and PhD degrees in photonic technology from the University of Malaya in 2001 and 2004. He is actively working on optical amplifiers, fiber optic sensors and fiber lasers. Sulaiman Wadi Harun is also an Associate

Fellow Member for both Academy of Sciences Malaysia and Malaysia Science Association. He is currently working as a full Professor in the Department of Electrical Engineering, University of Malaya, Malaysia.

Contents

**Preface IX**

Yury Dudchik

Chapter 1 **All-Optical Amplitude Multiplexing Through**

Chapter 3 **Effects of Dispersion Fiber on CWDM**

Chapter 5 **2 Terabit Transmission over Installed**

Chapter 6 **Advanced Modulation Formats and**

Paola Frascella and Andrew D. Ellis

Albert Gorshtein and Dan Sadot

Marcelo L. F. Abbade, Jorge D. Marconi, Eric A. M. Fagotto, Felipe R. Barbosa,

**Fiber Parametric Interaction Between Binary Signals 1**

André L. A. Costa, Iguatemi E. Fonseca and Edson Moschim

Chapter 2 **Multimode Passive Optical Network for LAN Application 35**  Elzbieta Beres-Pawlik, Grzegorz Budzyn and Grzegorz Lis

> **Directly Modulated System Performance 55**  Carmina del Río Campos and Paloma R. Horche

Chapter 4 **Design and Application of X-Ray Lens in the Form of Glass**

**SMF with Direct Detection Coherent WDM 95**

**MLSE Based Digital Signal Processing for**

Chapter 7 **Integration of Eco-Friendly POF Based Splitter and**

Chapter 8 **Secure Long-Distance Quantum Communication over Optical Fiber Quantum Channels 163**

Laszlo Gyongyosi and Sandor Imre

Mohammad Syuhaimi Ab-Rahman, Hadi Guna,

**Capillary Filled by a Set of Concave Epoxy Microlenses 77**

**100Gbit/sec Communication Through Optical Fibers 119**

**Optical Filter for Low-Cost WDM Network Solutions 145**

Mohd Hazwan Harun, Latifah Supian and Kasmiran Jumari

## Contents

### **Preface XI**

Laszlo Gyongyosi and Sandor Imre

X Contents

Chapter 17 **Fiber Optic Temperature Sensors 361**  S. W. Harun, M. Yasin, H. A. Rahman, H. Arof and H. Ahmad

## Preface

Since 1970, optical fiber and optical communication technologies have been rapidly developing, causing a technology revolution in the communication industries. Due to much lower attenuation and interference, optical fiber has many advantages over existing copper wire in long-distance and high-demand applications. The revolution in communication industries also significantly reduces the prices of optical components and stimulates the development of optical fiber sensors. This book is aimed at providing extensive overviews of the theoretical and experimental aspects of the current optical communication technologies as well as fiber optic devices. Chapter 1 describes a multilevel transmission technique based on optical amplitude multiplexing (OAM). Fiber nonlinear effects are used to perform OAM of binary signals into multi-amplitude signals. Recently, passive optical networks (PONs) have gained a lot of interest because they minimize the number of required optical transceivers, reduce the fiber optic infrastructure, and the need to power the intermediate network nodes. They are very suitable for fiber to home (FTTH) networks that support high speed internet and video on demand services. Chapter 2 describes several commercial applications for multimode PON structures designed and constructed to include N-equivalent nodes. Chapter 3 reviews the effects of dispersion in fiber on Course WDM (CWDM) directly modulated system performance. The performance of fibers relative to positive or negative dispersion characteristics is discussed for the case of directly modulated lasers. The effects of chirp and fiber nonlinearity in a directly modulated 2.5-Gb/s transmission system are also shown by simulation. It is observed that enhanced system performance, which uses a positive dispersion fiber, can be achieved if positive chromatic dispersion in the optical fiber is equalized by SPM, whereas laser transient chirp can be compensated using a negative dispersion fiber.

Chapter 4 reviews the design and application of X-ray lenses in the form of glass capillary filled by a set of concave epoxy micro-lenses. The fabrication and testing of compound refractive lenses (CRL) composed of micro-bubbles embedded in epoxy are discussed. The micro-bubble technique opens a new opportunity for designing lenses in the 8-9 keV range with focal lengths less than 30-40 mm. Chapter 5 demonstrates that direct detection CoWDM with Erbium-doped fiber amplifier (EDFA) amplification is only suitable for Terabit Ethernet transport over unrepeated spans of up to ~130 km. Raman amplification would allow for an increased system margin,

#### X Preface

where necessary. Experimental demonstration shows that a 124 km span transmission with Raman amplification left a Q-factor system margin of about 4 dB, which is consistent with theoretical expectations. Chapter 6 describes advanced modulation formats and MLSE based digital signal processing for 100Gbit/sec communication through optical fibers. Chapter 7 discusses polymer optical fiber (POF) splitters and WDM-POF network solutions.

Preface XI

**Dr Moh. Yasin,**

Indonesia

Malaysia

Malaysia

**Dr Hamzah Arof,**

Airlangga Univ. Surabaya,

**Professor Sulaiman W. Harun,** Department of Electrical Engineering, Faculty of Engineering, Univ. of Malaya,

Department of Electrical Engineering, Faculty of Engineering, Univ. of Malaya,

Department of Physics, Faculty of Science,

optic displacement sensor (FODS), lifetime measurements, microfiber loop resonator

(MLR), and stimulated brillouin scattering.

In today's communication networks, the widespread use of optical fiber and passive optical elements allows the use of quantum key distribution (QKD) in the current standard optical network infrastructure. Chapter 8 describes secure long-distance quantum communication over optical fiber quantum channels. A brief overview and description of the optical fiber-based quantum key distribution (QKD) protocols is presented. Also, the results of the information-theoretic security analysis of DPS (Differential Phase Shift) QKD protocol, which is designed for long-distance quantum communications between the quantum repeater nodes, are presented in this chapter. On the other hand, the wide range of applications of optical fibers have been continuously supported by their friendly integration with electronics. Chapter 9 explains an in-service line monitoring system for branched PON fibers. Chapter 10 describes nonlinear compensation using a multi-sub-band frequency-shaped backpropagation (FS-BP) approach. This allows flexible trade-off between performance and complexity as the number of steps and the number of sub-bands can be independently varied. Chapter 11 discusses optical performance analysis of single-mode fiber connections. Many SMF connection techniques, such as fusion splicing, mechanical splicing, and the use of optical connectors, are currently used in FTTH systems. The optical performance of SMF connections is reported for various cases.

The most common architecture used for establishing communication in optical networks is wavelength routing, where data is transmitted over all-optical WDM channels, called lightpaths, which may span multiple consecutive fibers. Wavelength routing could be carried out using optical add/drop multiplexers (OADMs), and optical cross-connectors (OXCs). Chapter 12 evaluates how a routing and wavelength assignment algorithm performs under various OXC node architectures. In chapter 13, it is demonstrated that one can use a semi-analytical receiver model to accurately estimate the performance of on-off-keyed (OOK) optical fiber communication systems, taking into account the impact of the choice of modulation format, arbitrarily polarized noise, and the receiver characteristics. Chapter 14 reviews wide area network (WAN) design topologies under redundancy constraints. This chapter presents a framework and algorithms suitable for addressing the design of minimal cost networks under connectivity constraints. Chapter 15 describes various optical transport architectures that perform unicast/multicast traffic grooming. Chapter 16 describes the main considerations for the design of digital electronic systems and how high speed optical links are handled in the electronic domain. Chapter 17 demonstrates various optical fiber temperature sensors based on four different techniques: intensity modulated fiber optic displacement sensor (FODS), lifetime measurements, microfiber loop resonator (MLR), and stimulated brillouin scattering.

X Preface

WDM-POF network solutions.

where necessary. Experimental demonstration shows that a 124 km span transmission with Raman amplification left a Q-factor system margin of about 4 dB, which is consistent with theoretical expectations. Chapter 6 describes advanced modulation formats and MLSE based digital signal processing for 100Gbit/sec communication through optical fibers. Chapter 7 discusses polymer optical fiber (POF) splitters and

In today's communication networks, the widespread use of optical fiber and passive optical elements allows the use of quantum key distribution (QKD) in the current standard optical network infrastructure. Chapter 8 describes secure long-distance quantum communication over optical fiber quantum channels. A brief overview and description of the optical fiber-based quantum key distribution (QKD) protocols is presented. Also, the results of the information-theoretic security analysis of DPS (Differential Phase Shift) QKD protocol, which is designed for long-distance quantum communications between the quantum repeater nodes, are presented in this chapter. On the other hand, the wide range of applications of optical fibers have been continuously supported by their friendly integration with electronics. Chapter 9 explains an in-service line monitoring system for branched PON fibers. Chapter 10 describes nonlinear compensation using a multi-sub-band frequency-shaped backpropagation (FS-BP) approach. This allows flexible trade-off between performance and complexity as the number of steps and the number of sub-bands can be independently varied. Chapter 11 discusses optical performance analysis of single-mode fiber connections. Many SMF connection techniques, such as fusion splicing, mechanical splicing, and the use of optical connectors, are currently used in FTTH systems. The

optical performance of SMF connections is reported for various cases.

The most common architecture used for establishing communication in optical networks is wavelength routing, where data is transmitted over all-optical WDM channels, called lightpaths, which may span multiple consecutive fibers. Wavelength routing could be carried out using optical add/drop multiplexers (OADMs), and optical cross-connectors (OXCs). Chapter 12 evaluates how a routing and wavelength assignment algorithm performs under various OXC node architectures. In chapter 13, it is demonstrated that one can use a semi-analytical receiver model to accurately estimate the performance of on-off-keyed (OOK) optical fiber communication systems, taking into account the impact of the choice of modulation format, arbitrarily polarized noise, and the receiver characteristics. Chapter 14 reviews wide area network (WAN) design topologies under redundancy constraints. This chapter presents a framework and algorithms suitable for addressing the design of minimal cost networks under connectivity constraints. Chapter 15 describes various optical transport architectures that perform unicast/multicast traffic grooming. Chapter 16 describes the main considerations for the design of digital electronic systems and how high speed optical links are handled in the electronic domain. Chapter 17 demonstrates various optical fiber temperature sensors based on four different techniques: intensity modulated fiber

#### **Dr Moh. Yasin,**

Department of Physics, Faculty of Science, Airlangga Univ. Surabaya, Indonesia

#### **Professor Sulaiman W. Harun,**

Department of Electrical Engineering, Faculty of Engineering, Univ. of Malaya, Malaysia

#### **Dr Hamzah Arof,**

Department of Electrical Engineering, Faculty of Engineering, Univ. of Malaya, Malaysia

**1**

*Brazil*

**All-Optical Amplitude Multiplexing**

**Between Binary Signals**

Marcelo L. F. Abbade1, Jorge D. Marconi2, Eric A. M. Fagotto1, Felipe R. Barbosa3,

*1Pontifícia Universidade Católica de Campinas*

*2Universidade Federal do ABC 3Universidade Estadual de Campinas 4Universidade Federal da Paraíba*

**Through Fiber Parametric Interaction**

André L. A. Costa3, Iguatemi E. Fonseca4 and Edson Moschim3

The high bandwidth and low attenuation provided by optical fibers has turned them into the most extensively deployed transmission medium in communication systems worldwide. This is especially the case for systems that utilize bit rates ranging from hundreds of Mbits/s to several Tb/s and whose span extends from a few tens of kilometers to intercontinental scales. In fact, global networking with the present speed and quality could hardly exist without fibers, which transport more information than all the other

The interest in optical fibers goes far beyond their valuable characteristics for signal propagation. In particular, fiber nonlinearities have been widely considered for the implementation of several all-optical devices. For example, wavelength converters based on cross-phase modulation (XPM) (Olsson et al., 2000) and fiber four-wave mixing (FWM) (Inoue & Toba, 1992) have been investigated. Dispersion compensators and all-optical regenerators FWM have also been implemented (Chavez Boggio et al., 2004a), as well as wide-band tunable amplifiers, known as fiber optic parametric amplifiers (FOPA), relying on third-order parametric processes with one (Hansryd & Andrekson, 2001) and two highpower pumps have been demonstrated (Chavez Boggio et al., 2005b). Recent work (Jamshidifar et al., 2010) shows that fiber tunable filters and demultiplexers can be achieved

Fiber-based nonlinear devices may also be used for all-optical signal processing. In this area, there is a special interest on techniques that provide conversion between different modulation formats. In fact, recent works deal with this subject and propose ways of performing analog to digital (Brzozowski & Sargent, 2001), digital to analog (Oda & Maruta, 2006), non-return-to-zero (NRZ) to return-to-zero (RZ) (Mishina et al., 2007), multilevel to

transmission media commercially used today combined (Ramaswamy, 2010).

through parametric interaction in specially designed optical fibers.

**1. Introduction**

## **All-Optical Amplitude Multiplexing Through Fiber Parametric Interaction Between Binary Signals**

Marcelo L. F. Abbade1, Jorge D. Marconi2, Eric A. M. Fagotto1, Felipe R. Barbosa3, André L. A. Costa3, Iguatemi E. Fonseca4 and Edson Moschim3 *1Pontifícia Universidade Católica de Campinas 2Universidade Federal do ABC 3Universidade Estadual de Campinas 4Universidade Federal da Paraíba Brazil*

## **1. Introduction**

The high bandwidth and low attenuation provided by optical fibers has turned them into the most extensively deployed transmission medium in communication systems worldwide. This is especially the case for systems that utilize bit rates ranging from hundreds of Mbits/s to several Tb/s and whose span extends from a few tens of kilometers to intercontinental scales. In fact, global networking with the present speed and quality could hardly exist without fibers, which transport more information than all the other transmission media commercially used today combined (Ramaswamy, 2010).

The interest in optical fibers goes far beyond their valuable characteristics for signal propagation. In particular, fiber nonlinearities have been widely considered for the implementation of several all-optical devices. For example, wavelength converters based on cross-phase modulation (XPM) (Olsson et al., 2000) and fiber four-wave mixing (FWM) (Inoue & Toba, 1992) have been investigated. Dispersion compensators and all-optical regenerators FWM have also been implemented (Chavez Boggio et al., 2004a), as well as wide-band tunable amplifiers, known as fiber optic parametric amplifiers (FOPA), relying on third-order parametric processes with one (Hansryd & Andrekson, 2001) and two highpower pumps have been demonstrated (Chavez Boggio et al., 2005b). Recent work (Jamshidifar et al., 2010) shows that fiber tunable filters and demultiplexers can be achieved through parametric interaction in specially designed optical fibers.

Fiber-based nonlinear devices may also be used for all-optical signal processing. In this area, there is a special interest on techniques that provide conversion between different modulation formats. In fact, recent works deal with this subject and propose ways of performing analog to digital (Brzozowski & Sargent, 2001), digital to analog (Oda & Maruta, 2006), non-return-to-zero (NRZ) to return-to-zero (RZ) (Mishina et al., 2007), multilevel to

All-Optical Amplitude Multiplexing Through

where

*L*, and

where A = [*J*0 (

singlemode fibers.

where (*n*/*c*)2 = (

0

Considering a typical value

the fields can be written as:

amplitude of the electric field,

*a* is the core radius, and 222

core and the cladding, respectively (Agrawal, 2001).

longitudinal components of the electric fields are of the order of

(

(*x*,*y*) is the transverse distribution of the electric field:

*Ta*)]-1 and B = [K0 (T*a*)]-1, 22 2

magnetic material without free charges and currents,

2

that is related with the magnetic induction field *B* as *B H*

2

Fiber Parametric Interaction Between Binary Signals 3

product of the (3) tensor and the *EEE* tensor leads a simple vector. Furthermore, considering that silica glass is an isotropic material, the number of independent elements of the (3) tensor is reduced to three independent elements by symmetry (Buck, 2005; Butcher & Cotter, 1990). In addition, if the operating frequency range is far from resonances, then the number of independent elements is finally reduced to one (Buck, 2005). The "cubic" term of Eq. (1) is responsible for several nonlinear effects in optical fibers such as self-phase modulation (SPM), XPM, and FWM. It will be seen that, under proper conditions, FWM can be used to amplify a weak signal that propagates through an optical fiber along with a strong signal. To show that, let us start with the wave equation (Buck, 2005) for silica, a non-

> 0 0 0 2 2 *H EP E*

Assuming that *J Pt* / , then the total polarization can be written as a sum of two terms, the linear and the nonlinear polarizations: *PP P L NL* . Therefore, Eq. (2) can be written as

2 2 2

vacuum. In the case of single mode fibers that are used in parametric devices, we have that

are ~20 times smaller than the transversal components; therefore, they can be neglected in most practical applications. Then, considering that the fiber propagation is along the *z*-axis,

> 1 (( ) ) ˆ (, ) [ () . .] 2 *L L i zt E x xy A ze L L c c*

where *c.c.* stands for the complex conjugate of the previous term, *AL*(*z*) is the complex

0 0 ( ) (,) ( )

to the fundamental mode (called *HE*11 or *LP*01) which is the only one propagating in

*x y B K a*

*T T A J a*

*T core n k*0

*x y* . Here *J*0 and *K*0 are the Bessel functions corresponding

, (4)

*L*) is the propagation constant for the angular frequency

, (5)

 , 22 2 *T n k cladding* 0

, 0*k* 2 /

 ,

= (*n*core – *n*cladding)/*n*cladding << 1, where *n*core and *n*cladding are the refractive indexes of the

2 2 0 2 *n PNL E ct t*

*t t t*

0 is the vacuum magnetic permeability, and *H* is the magnetic field (Jackson, 1998)

2 2

0(1 + (1))), being *n* the refractive index and c the speed of light in

~ 0.003, for weakly guiding fibers (Gloge, 1971), the

. (2)

0 (then *B H*

, (3)

1/2, which means that they

0 ).

binary (Fagotto & Abbade, 2010) and binary to multilevel conversions (Zhou et al., 2006; Lu & Miyazaki, 1997), among others.

A strong motivation for pursuing such research is that different kinds of optical networks (long-haul, optical packet switching, access and so on) may coexist and need to exchange information with one another (Mishina et al., 2007). Since each of them may present different physical and logical characteristics, signals within the boundaries of each optical network are subject to different types of impairments. Therefore, a modulation format used to minimize the bit error rate (BER) of signals within a given network domain may not be appropriate for other domains. Consequently, all-optical devices that provide modulation format conversion capabilities may be highly attractive to play a major role at the interface between different optical networks.

In this chapter, we focus on four techniques that use fiber nonlinear effects to perform optical amplitude multiplexing (OAM) of binary signals into multi-amplitude ones. We begin by reviewing the fundamentals of FWM and parametric amplification (PA) in Section 2. Then, in Section 3, we discuss two techniques that use FWM (Abbade et al., 2005; Abbade et al., 2006a; Abbade et al., 2006b) and PA (Abbade et al., 2010a; Abbade et al., 2010b; Marconi et al., 2011) to convert two 2-ASK signals into a quaternary amplitude-shift keying (4-ASK) one; named, respectively, OAM-4F and OAM-4P. In Section 4, utilization of FWM and PA (Abbade et al., 2011) to convert two binary signals into a ternary amplitude-shift keying (3-ASK) is approached. Such techniques are, respectively, termed OAM-3F and OAM-3P. It should be noted that OAM-3F is an innovation, presented for the first time in this work. Advantages of multi-amplitude modulation formats encompass higher tolerance to degradations caused by chromatic dispersion and the possibility of transmitting simultaneously two signals within the same optical bandwidth. Applications and a detailed comparison among the four techniques are presented in Section 5. Finally, conclusions are drawn in Section 6.

## **2. Theory of parametric interactions in optical fibers**

An external electric field *E* applied to an optical fiber will cause an induced polarization *P* in the medium that will depend on its electrical susceptibility . However, such a dependence will not only rely on the first-order susceptibility (1) , but also on higher-order terms. Processes relying on (2) and (3) are known, respectively, as second- and thirdorder parametric processes.

For an isotropic medium, the second order susceptibility (2) is equal to zero (in dipole approximation), which means that the term proportional to 2 *E* vanishes. Therefore, only the contribution from the third order susceptibility (3), proportional to 3 *E* will remain. Thus, the polarization can be expressed as,

$$
\overline{P} = \varepsilon\_0(\mathcal{X}^{(1)}\overline{E} + \mathcal{X}^{(3)}\overline{E}^3),
\tag{1}
$$

where 0 is the vacuum electric permittivity.

At this point it is important to comment on two aspects of Eq. (1). First, the notation 3 *E* is used as a simple way of writing the triple external product *EEE* , which results in a third rank tensor with 27 elements. Second, the third order susceptibility (3) is a fourth rank tensor which contains 81 independent elements, but it should be noted that the inner

binary (Fagotto & Abbade, 2010) and binary to multilevel conversions (Zhou et al., 2006; Lu

A strong motivation for pursuing such research is that different kinds of optical networks (long-haul, optical packet switching, access and so on) may coexist and need to exchange information with one another (Mishina et al., 2007). Since each of them may present different physical and logical characteristics, signals within the boundaries of each optical network are subject to different types of impairments. Therefore, a modulation format used to minimize the bit error rate (BER) of signals within a given network domain may not be appropriate for other domains. Consequently, all-optical devices that provide modulation format conversion capabilities may be highly attractive to play a major role at the interface

In this chapter, we focus on four techniques that use fiber nonlinear effects to perform optical amplitude multiplexing (OAM) of binary signals into multi-amplitude ones. We begin by reviewing the fundamentals of FWM and parametric amplification (PA) in Section 2. Then, in Section 3, we discuss two techniques that use FWM (Abbade et al., 2005; Abbade et al., 2006a; Abbade et al., 2006b) and PA (Abbade et al., 2010a; Abbade et al., 2010b; Marconi et al., 2011) to convert two 2-ASK signals into a quaternary amplitude-shift keying (4-ASK) one; named, respectively, OAM-4F and OAM-4P. In Section 4, utilization of FWM and PA (Abbade et al., 2011) to convert two binary signals into a ternary amplitude-shift keying (3-ASK) is approached. Such techniques are, respectively, termed OAM-3F and OAM-3P. It should be noted that OAM-3F is an innovation, presented for the first time in this work. Advantages of multi-amplitude modulation formats encompass higher tolerance to degradations caused by chromatic dispersion and the possibility of transmitting simultaneously two signals within the same optical bandwidth. Applications and a detailed comparison among the four techniques

An external electric field *E* applied to an optical fiber will cause an induced polarization *P*

For an isotropic medium, the second order susceptibility (2) is equal to zero (in dipole

3 (1) (3) 0 *P EE*

At this point it is important to comment on two aspects of Eq. (1). First, the notation 3

used as a simple way of writing the triple external product *EEE* , which results in a third rank tensor with 27 elements. Second, the third order susceptibility (3) is a fourth rank tensor which contains 81 independent elements, but it should be noted that the inner

  are known, respectively, as second- and third-

( ) , (1)

. However, such a

*E* will remain. Thus,

*E* is

, but also on higher-order

*E* vanishes. Therefore, only the

are presented in Section 5. Finally, conclusions are drawn in Section 6.

in the medium that will depend on its electrical susceptibility

 and (3)

contribution from the third order susceptibility (3), proportional to 3

dependence will not only rely on the first-order susceptibility (1)

approximation), which means that the term proportional to 2

is the vacuum electric permittivity.

**2. Theory of parametric interactions in optical fibers**

& Miyazaki, 1997), among others.

between different optical networks.

terms. Processes relying on (2)

the polarization can be expressed as,

order parametric processes.

where 0

product of the (3) tensor and the *EEE* tensor leads a simple vector. Furthermore, considering that silica glass is an isotropic material, the number of independent elements of the (3) tensor is reduced to three independent elements by symmetry (Buck, 2005; Butcher & Cotter, 1990). In addition, if the operating frequency range is far from resonances, then the number of independent elements is finally reduced to one (Buck, 2005). The "cubic" term of Eq. (1) is responsible for several nonlinear effects in optical fibers such as self-phase modulation (SPM), XPM, and FWM. It will be seen that, under proper conditions, FWM can be used to amplify a weak signal that propagates through an optical fiber along with a strong signal. To show that, let us start with the wave equation (Buck, 2005) for silica, a nonmagnetic material without free charges and currents,

$$
\nabla^2 \overline{E} = \mu\_0 \frac{\partial \overline{\nabla} \times \overline{H}}{\partial t} = \mu\_0 \left( \varepsilon\_0 \frac{\partial^2 \overline{E}}{\partial t^2} + \frac{\partial^2 \overline{P}}{\partial t^2} \right). \tag{2}
$$

where 0 is the vacuum magnetic permeability, and *H* is the magnetic field (Jackson, 1998) that is related with the magnetic induction field *B* as *B H* 0 (then *B H* 0 ). Assuming that *J Pt* / , then the total polarization can be written as a sum of two terms, the linear and the nonlinear polarizations: *PP P L NL* . Therefore, Eq. (2) can be written as

$$
\left(\nabla^2 - \frac{n^2}{c^2} \frac{\partial^2}{\partial t^2}\right) \overline{E} = \mu\_0 \frac{\partial^2 \overline{P\_{NL}}}{\partial t^2} \,, \tag{3}
$$

where (*n*/*c*)2 = (00(1 + (1))), being *n* the refractive index and c the speed of light in vacuum. In the case of single mode fibers that are used in parametric devices, we have that = (*n*core – *n*cladding)/*n*cladding << 1, where *n*core and *n*cladding are the refractive indexes of the core and the cladding, respectively (Agrawal, 2001).

Considering a typical value ~ 0.003, for weakly guiding fibers (Gloge, 1971), the longitudinal components of the electric fields are of the order of 1/2, which means that they are ~20 times smaller than the transversal components; therefore, they can be neglected in most practical applications. Then, considering that the fiber propagation is along the *z*-axis, the fields can be written as:

$$\overline{E}\_L = \hat{\mathbf{x}}\psi(\mathbf{x}, y)\frac{1}{2}[A\_L(\mathbf{z})e^{-i(\beta\{\alpha\_l\}z - \alpha\_l t)} + \text{c.c.}]\,\tag{4}$$

where *c.c.* stands for the complex conjugate of the previous term, *AL*(*z*) is the complex amplitude of the electric field, (*L*) is the propagation constant for the angular frequency *L*, and (*x*,*y*) is the transverse distribution of the electric field:

$$\varphi(\mathbf{x}, y) \propto \begin{cases} A & f\_0(\kappa\_T \rho) \\ B \ K\_0(\mathbf{y}\_T \rho) & \rho > a \end{cases} \tag{5}$$

where A = [*J*0 (*Ta*)]-1 and B = [K0 (T*a*)]-1, 22 2 *T core n k*0 , 22 2 *T n k cladding* 0 , 0*k* 2 / , *a* is the core radius, and 222 *x y* . Here *J*0 and *K*0 are the Bessel functions corresponding to the fundamental mode (called *HE*11 or *LP*01) which is the only one propagating in singlemode fibers.

All-Optical Amplitude Multiplexing Through

 *j*= ( ) *j 2*/*j*, with *j* = *2c*/*j,*

 *s id P* , and *n cA* 2 / *eff*

power, which justifies the approximation.

Under such conditions Eqs.(10)-(12) may be written as,

*dz*

*dz*

where the symbol (\*) stands for the complex conjugate,

*dz*

*dz*

*dz*

( ) ( ) 2( )

terms of SPM, XPM and FWM.

 or, considering that

wavelength distance (

be neglected.

Fiber Parametric Interaction Between Binary Signals 5

*j*

set of three coupled equations for the three electric field amplitudes,

*j j*

*j*) as compared to the envelope magnitude (Buck, 2005). Using this

, (9)

is the fiber nonlinear coefficient.  Note that the right-hand side of Eqs. (11)-(13) includes the

The exact solutions of Eqs. (10)-(12) involve Jacobian elliptical functions as shown in (Chen, 1989). Here we follow an approximate solution that allows us to obtain a simple expression for the parametric gain. This approximation considers that the intensity of the pump is much higher than that of the signal and the idler. Therefore, the energy transferred from the pump to the signal (and the idler) can be considered negligible. For instance, if the ratio between the pump power and the signal power at the fiber input is ~ 104 -105, and the signal gain is ~20-25 dB at the fiber output, the signal (idler) power is still less than 1% of the pump

> *P p P*

\* 2 [2 exp( )] *S p S id P dA i PA A A i z*

\* 2 [2 exp( )] *id p id S id dA i PA A A i z*

*dA i PA dz*

the nonlinear refractive index which is related to (3) as (3) 2

As previously mentioned, the nonlinear process responsible for parametric amplification is the FWM. To show this let us assume that the refractive index can be written as the sum of a linear term and a nonlinear contribution, *n n In* 0 2 , where *n*0 is the linear part of the refractive index and *n*2 is the nonlinear refractive index (Boyd, 2008). The nonlinear contribution is proportional to the optical irradiance *I* (in the SI units system) (Boyd, 2008). Then, when two waves at angular frequencies *1* and *2* are launched together into an optical fiber, the refractive index will be modulated with a frequency (*2* - *1*). Now, if a third wave at frequency 3 is coupled along, a new wave at frequency *4* = *3* ± (*2* - *1*) will be generated. This new wave is called idler. The relation *4* = *3* ± (*2* - *1*) means that *4* + *1* = *3* + *2* and that *4* + *2* = *3* + *1*.

It is important to mention that when three waves are launched into the same fiber, a total of nine new frequencies can be in fact generated if all the combinations are taken into account (Hansryd et al., 2002). For instance, frequencies such as *4* = 2*2* - *3* or *4* = *3* - *2* + *1* are also possible. However, not all these frequencies are generated with the same efficiency. Generally calculations reckon only highly efficient processes and neglect the others. For instance, if we consider the case where *1* < *3* = *2*, we have that 2*2* = *4* + *1*; this process is a degenerate case of FWM that brings about parametric amplification when the wave at 2 is a strong pump-signal and 1 is a weak signal to be amplified. As a result, a new wave, an amplified copy of the signal at *1*, will be generated at 4.

In order to standardize the notation, we shall denote the angular frequencies for the pump, the signal and the idler as P, S and id, respectively. Following this notation, the total electric field can be written as

$$\begin{split} \overline{E} &= \overline{E}p + \overline{E}s + \overline{E}id = \hat{\mathbf{x}} \,\nu(\mathbf{x}, \mathbf{y}) \frac{1}{2} [A\_P(z)e^{-i(\beta(\alpha\_\flat)z - \alpha\_\flat t)} + \\ &+ A\_S(z)e^{-i(\beta(\alpha\_\flat)z - \alpha\_\flat t)} + \\ &+ A\_{id}(z)e^{-i(\beta(\alpha\_{\acute{u}})z - \alpha\_{\acute{u}}t)} \Big] + c.c. \end{split} \tag{6}$$

where all the waves are supposed to have the same mode profile *ψ*(*x*,*y*) and the same polarization on the *x*-axis. When this total electric field is included in Eq. (3), the Laplacian leads to the following electric fields (pump, signal, and idler)

$$\begin{split} \frac{\partial^2 \overline{E\_j}}{\partial z^2} &= \hat{\mathbf{x}} \nu(\mathbf{x}, y) \frac{1}{2} [\frac{\hat{\sigma}^2 A\_j(z)}{\hat{\sigma} z^2} e^{-i(\beta(\boldsymbol{\alpha}\_j)z - \boldsymbol{\alpha}\_j t)} - \\ &- 2i\beta(\boldsymbol{\alpha}\_j) \frac{A\_j(z)}{\hat{\sigma} z} e^{-i(\beta(\boldsymbol{\alpha}\_j)z - \boldsymbol{\alpha}\_j t)} - \\ &- \beta^2 \left( \boldsymbol{\alpha}\_j \right) A\_j(z) e^{-i(\beta(\boldsymbol{\alpha}\_j)z - \boldsymbol{\alpha}\_j t)} \big] + c.c. \end{split} \tag{7}$$

where *j* = *P*, *S* or *id*.

The slowly varying envelope approximation is introduced at this point, and is given by

$$\left|\frac{\partial^2 A\_j}{\partial z^2}\right| << \left|\beta\_j \frac{\partial A\_j}{\partial z}\right| \tag{8}$$

As previously mentioned, the nonlinear process responsible for parametric amplification is the FWM. To show this let us assume that the refractive index can be written as the sum of a linear term and a nonlinear contribution, *n n In* 0 2 , where *n*0 is the linear part of the refractive index and *n*2 is the nonlinear refractive index (Boyd, 2008). The nonlinear contribution is proportional to the optical irradiance *I* (in the SI units system) (Boyd, 2008).

It is important to mention that when three waves are launched into the same fiber, a total of nine new frequencies can be in fact generated if all the combinations are taken into account

also possible. However, not all these frequencies are generated with the same efficiency. Generally calculations reckon only highly efficient processes and neglect the others. For

is a degenerate case of FWM that brings about parametric amplification when the wave at 2 is a strong pump-signal and 1 is a weak signal to be amplified. As a result, a new wave, an

In order to standardize the notation, we shall denote the angular frequencies for the pump,

(( ) ) (( ) )

*i zt*

*S S id id*

1 ˆ (, ) [ () 2

*i zt P S id P*

( ) ] ..

*i zt*

*A ze c c*

where all the waves are supposed to have the same mode profile *ψ*(*x*,*y*) and the same polarization on the *x*-axis. When this total electric field is included in Eq. (3), the Laplacian

*j j i zt*

2 (( ) )

The slowly varying envelope approximation is introduced at this point, and is given by

*1*, will be generated at

*1* < *3* =

( )

*A ze*

*S*

*E E E E x xy A ze*

*id*

2 2

2 2

*z z*

*E A z x xy e*

( ) 2()

*j*

*j j*

2 2

1 ( ) ˆ (,) [ 2

*A z i e z*

  *1* and

> *4* = *3* ± (*2* -

*4* = 2*2* - *3* or *4* = *3* - *2* + *1* are

*2*, we have that 2

4. *2* are launched together into an

*1*). Now, if a

*1*; this process

, (6)

(7)

*4* +

*1*) means that

*2* -

*4* = *3* ± (*2* - *1*) will

*2* = *4* +

id, respectively. Following this notation, the total

(( ) )

(( ) )

*j j*

, (8)

(( ) )

*j j*

( ) ] ..

*i zt*

*A ze c c*

*j j j A A z z*

*j i zt*

*j j*

*P P*

Then, when two waves at angular frequencies

*4* + *2* = *3* + *1*.

instance, if we consider the case where

*2* and that

amplified copy of the signal at

the signal and the idler as

where *j* = *P*, *S* or *id*.

electric field can be written as

*1* = *3* +

be generated. This new wave is called idler. The relation

(Hansryd et al., 2002). For instance, frequencies such as

P, S and

leads to the following electric fields (pump, signal, and idler)

optical fiber, the refractive index will be modulated with a frequency (

third wave at frequency 3 is coupled along, a new wave at frequency

or, considering that *j*= ( ) *j 2*/*j*, with *j* = *2c*/*j,*

$$\left| \mathcal{A}\_j \frac{\partial}{\partial z} \left( \frac{\partial A\_j}{\partial z} \right) \right| << \left| \frac{\partial A\_j}{\partial z} \right| \,, \tag{9}$$

which means that the slopes of the envelope fields do not vary significantly along a wavelength distance (*j*) as compared to the envelope magnitude (Buck, 2005). Using this approximation, valid for most practical cases, the term with the second derivative of *z* can be neglected.

Considering all these conditions and neglecting fiber attenuation, it is possible to obtain a set of three coupled equations for the three electric field amplitudes,

$$\frac{dA\_P}{dz} = i\gamma \left\{ \left| A\_P \right|^2 + 2\left( \left| A\_S \right|^2 + \left| A\_{id} \right|^2 \right) \right\} A\_P + 2A\_S A\_{id} A\_P^\* \exp(i\Delta\beta\cdot\mathbf{j})\,\tag{10}$$

$$\frac{dA\_S}{dz} = i\gamma \left[ \left| A\_S \right|^2 + 2\left( \left| A\_{id} \right|^2 + \left| A\_P \right|^2 \right) \right] A\_S + A\_{id}^\* A\_P^2 \exp(-i\Delta\beta \cdot \mathbf{z}) \, \tag{11}$$

$$\frac{dA\_{id}}{dz} = i\gamma \{ \left| A\_{id} \right|^2 + 2\left( \left| A\_S \right|^2 + \left| A\_P \right|^2 \right) \} A\_{id} + A\_S \, ^\*A\_P \, ^2 \exp\{-i\Lambda\beta\beta\} \, . \tag{12}$$

where the symbol (\*) stands for the complex conjugate, is the linear phase mismatch ( ) ( ) 2( ) *s id P* , and *n cA* 2 / *eff* is the fiber nonlinear coefficient. Here *n*2 is

the nonlinear refractive index which is related to (3) as (3) 2 2 0 0 *n* 3 /4 *cn* , and *Aeff* is the effective area (Agrawal, 2001). Note that the right-hand side of Eqs. (11)-(13) includes the terms of SPM, XPM and FWM.

The exact solutions of Eqs. (10)-(12) involve Jacobian elliptical functions as shown in (Chen, 1989). Here we follow an approximate solution that allows us to obtain a simple expression for the parametric gain. This approximation considers that the intensity of the pump is much higher than that of the signal and the idler. Therefore, the energy transferred from the pump to the signal (and the idler) can be considered negligible. For instance, if the ratio between the pump power and the signal power at the fiber input is ~ 104 -105, and the signal gain is ~20-25 dB at the fiber output, the signal (idler) power is still less than 1% of the pump power, which justifies the approximation.

Under such conditions Eqs.(10)-(12) may be written as,

$$\frac{dA\_P}{dz} = i\gamma \left(P\_p A\_P\right)\_{\prime} \tag{13}$$

$$\frac{dA\_S}{dz} = i\gamma \left[ 2P\_p A\_S + A\_{id}^\* A\_P^2 \exp(-i\Lambda\beta\,\,\dot{z}) \right],\tag{14}$$

$$\frac{dA\_{id}}{dz} = \text{i}\gamma \left[ 2P\_p A\_{id} + A\_S \stackrel{\*}{\!\!\/} A\_{id} \stackrel{\*}{\!\!\/} \exp(-i\Delta\beta \not\!\/)\right].\tag{15}$$

All-Optical Amplitude Multiplexing Through

all these results in Eq.(17) we have that

,+ = 2 *1,2* – *2,1*.

*P*, *S*, and *id* to *1*, *2*, and

>

frequencies change from

waves at frequencies

**3.1.1 Theory**

be rewritten as:

If the channels at

1 and

signals at

2, − , and

just considered implies that

Fiber Parametric Interaction Between Binary Signals 7

, *x i L*

2 22

**3. All-optical generation of quaternary amplitude-shift keying signals**

is presented in Section 3.1 and OAM-4P is approached in Section 3.2.

**3.1 Optical amplitude multiplexing through fiber four-wave mixing**

that favors the occurrence of FWM. When the fiber attenuation coefficient

*P PP L*

*1* and *2* are:

where *P*1, *P*2, *P*-, and *P*+ are the respective optical powers of the channels at frequencies

wavelength and intensity-dependent FWM generation efficiency, well described in the

2 sin( ) 4 *L L*

*P-,+* is null whenever one of these input channels transmits a 0-bit. Here, however, we assume that these channels are codified by a binary amplitude-shift keying (2-ASK) scheme, where the 0-bit powers of are intentionally offset. In this case, the extinction ratios (ER)

*L*

2 2 , 1,2 2,1

+, *L* is the fiber length,

literature (Mussot et al., 2007), which is given by:

*1* and

corresponding to the channels at

sin( ) 4 *L P PP L*

The notation *P*−,+ refers to the power of the two principal FWM processes that generate

This section presents two all-optical techniques for multiplexing two binary ASK signals (ASK-2), traveling at different carrier wavelengths, in a single 4-ASK signal. In the first case, OAM-4F, the four levels of the quaternary pattern are obtained when the two binary signals, which have similar optical power, interact through FWM. The theoretical calculations that allow estimation of the power of the quaternary levels are developed from Eq. (27). The second approach, OAM-4P, used to generate the single 4-ASK signal from two binary ASK signals is based on PA. In this case, one of the signals is a strong optical signal that acts as a pump, with the unusual characteristic of being modulated by binary information. OAM-4F

The diagram shown in Fig. 1 illustrates the principle of OAM-4F. Two co-polarized input

2 are coupled into a fiber, where they co-propagate through a medium

2

2

*L*

is the fiber nonlinear coefficient, and

. (22)

*2* are codified with ideal on-off keying (OOK) modulation, then

1 exp exp

, 1,2 2,1

,+, respectively. The extreme case

≠ 0, Eq. (20) can

is the

1,

, (21)

. Introducing

, and then sinh( ) 2 sin *xi L*

. (20)

2

*L*

where *PP* = 2 *AP* is the pump power. These equations, which are valid for an ideal fiber with attenuation coefficient = 0, have an analytical solution for the signal gain and for the idler conversion efficiencies as,

$$\frac{P\_S}{P\_S(0)} = 1 + \left(\frac{x\_0 \sinh(\mathbf{x})}{\mathbf{x}}\right)^2,\tag{16}$$

$$\frac{P\_{id}}{P\_S(0)} = \left(\frac{\pi\_0 \sinh(\chi)}{\chi}\right)^2,\tag{17}$$

where *PS* = 2 *AS* is the signal power at the fiber output, *PS* (0) is the signal power at the fiber input, *Pid* = 2 *Aid* is the idler power at the fiber output, *L* is the fiber length, 0 *P x PL* , 2 0 1 ( /2 ) *T P x x P* , and 2 *T PP* .

It should be noted that even when ≠ 0, previous equations are good approximations if the pump power *PP* is replaced by 0 1 ( ) (0) 1 / *L L P P z dz P e L PP P L* . The phase mismatch can be calculated by expanding () in Taylor series around an arbitrary frequency *t* as follows:

$$\mathcal{J}(\boldsymbol{\alpha}) = \mathcal{J}(\boldsymbol{\alpha}\_{l}) + \left(\frac{\partial \mathcal{J}}{\partial \boldsymbol{\alpha}}\right)\_{\boldsymbol{\alpha} = \boldsymbol{\alpha}\_{l}} \left\{ \boldsymbol{\alpha} - \boldsymbol{\alpha}\_{l} \right\} + \frac{1}{2} \left(\frac{\hat{\boldsymbol{\mathcal{E}}}^{2} \mathcal{J}}{\hat{\boldsymbol{\mathcal{E}}} \boldsymbol{\alpha}^{2}}\right)\_{\boldsymbol{\alpha} = \boldsymbol{\alpha}\_{l}} \left\{ \boldsymbol{\alpha} - \boldsymbol{\alpha}\_{l} \right\}^{2} +$$

$$+ \frac{1}{6} \left(\frac{\hat{\boldsymbol{\mathcal{E}}}^{3} \mathcal{J}}{\hat{\boldsymbol{\mathcal{E}}} \boldsymbol{\alpha}^{3}}\right)\_{\boldsymbol{\alpha} = \boldsymbol{\alpha}\_{l}} \left( \boldsymbol{\alpha} - \boldsymbol{\alpha}\_{l} \right)^{3} + \frac{1}{24} \left( \frac{\hat{\boldsymbol{\mathcal{E}}}^{4} \mathcal{J}}{\hat{\boldsymbol{\mathcal{E}}} \boldsymbol{\alpha}^{4}}\right)\_{\boldsymbol{\alpha} = \boldsymbol{\alpha}\_{l}} \left( \boldsymbol{\alpha} - \boldsymbol{\alpha}\_{l} \right)^{4} + \dots \dots \tag{18}$$

Keeping terms up to the fourth order and taking *t* = *P* then,

$$
\Delta\beta(\alpha) = \beta\_2(\alpha\_\text{P})(\alpha - \alpha\_\text{P})^2 + \frac{\beta\_4(\alpha\_\text{P})}{12}(\alpha - \alpha\_\text{P})^4 \,. \tag{19}
$$

Within the spectral region where the parameter *x* is real, the parametric gain is maximum when T = 0 (*x* = *x*0), and its value is *Gmax* = 1 + *sinh*2(*x*0). On the other hand, the gain has a local minimum (within the region of interest) when = 0, and its value is *Gmin* = 1 + 2 0*x* (Chavez Boggio et al., 2005a). In other words, the parametric gain will be high if T is small. This means that the pump must be tuned at some frequency within the fiber anomalous dispersion region, that is, P < 0, with 0 = *2πc*/0, 0 the fiber zero dispersion wavelength and *c* is the vacuum light speed. If the approximation 4 ~ 0 is valid, then the gain bandwidth can be roughly written as 2 *P P* 2 / *P* .

In the extreme case of *Pp* << and *PP* ~ *PS* >> *Pid*, Eq. (17) gives the mixing condition without amplification (Stolen & Bjorkholm, 1982). Here we change our notation and designate, the pump, the signal, and the idler powers as *P1*, *P2*, and *P*−,+ , and the angular frequencies change from *P*, *S*, and *id* to *1*, *2*, and ,+, respectively. The extreme case just considered implies that , *x i L* , and then sinh( ) 2 sin *xi L* . Introducing all these results in Eq.(17) we have that

$$P\_{-,+} = 4P\_{1,2}^2 P\_{2,1} \mathcal{V}^2 L^2 \left(\frac{\sin(\Delta \mathcal{J} L)}{\Delta \mathcal{J} L}\right)^2. \tag{20}$$

The notation *P*−,+ refers to the power of the two principal FWM processes that generate waves at frequencies ,+ = 2 *1,2* – *2,1*.

## **3. All-optical generation of quaternary amplitude-shift keying signals**

This section presents two all-optical techniques for multiplexing two binary ASK signals (ASK-2), traveling at different carrier wavelengths, in a single 4-ASK signal. In the first case, OAM-4F, the four levels of the quaternary pattern are obtained when the two binary signals, which have similar optical power, interact through FWM. The theoretical calculations that allow estimation of the power of the quaternary levels are developed from Eq. (27). The second approach, OAM-4P, used to generate the single 4-ASK signal from two binary ASK signals is based on PA. In this case, one of the signals is a strong optical signal that acts as a pump, with the unusual characteristic of being modulated by binary information. OAM-4F is presented in Section 3.1 and OAM-4P is approached in Section 3.2.

#### **3.1 Optical amplitude multiplexing through fiber four-wave mixing**

#### **3.1.1 Theory**

6 Optical Fiber Communications and Devices

0 sinh( ) 1

0 sinh( )

2 *AS* is the signal power at the fiber output, *PS* (0) is the signal power at the fiber

2 *Aid* is the idler power at the fiber output, *L* is the fiber length, 0 *P x PL*

1 ( ) (0) 1 /

2

3 4

*t t*

*t* = *P* then,

*P P P*

= 0 (*x* = *x*0), and its value is *Gmax* = 1 + *sinh*2(*x*0). On the other hand, the gain has a

0,

*P P* 2 /

2

2 4 4

and *PP* ~ *PS* >> *Pid*, Eq. (17) gives the mixing condition

 *P* .

 These equations, which are valid for an ideal fiber

= 0, have an analytical solution for the signal gain and for the

, (16)

, (17)

≠ 0, previous equations are good approximations if the

. The phase mismatch

2

) in Taylor series around an arbitrary frequency

  ,

> *t* as

> > 0*x*

is small.

(18)

. (19)

T

= 0, and its value is *Gmin* = 1 + 2

0 the fiber zero dispersion wavelength

4 ~ 0 is valid, then the gain

2

2

where *PP* =

where *PS* =

input, *Pid* =

follows:

when T

dispersion region, that is,

In the extreme case of *Pp* <<

0 1 ( /2 ) *T P x x*

with attenuation coefficient

idler conversion efficiencies as,

2

 *P* , and 2

pump power *PP* is replaced by

can be calculated by expanding

Keeping terms up to the fourth order and taking

local minimum (within the region of interest) when

P < 0, with 0 = *2πc*/

and *c* is the vacuum light speed. If the approximation

bandwidth can be roughly written as 2

It should be noted that even when

(0) *S S*

> (0) *id S*

 *T PP* .

0

1 () ( ) ( ) ( ) 2 *t t tt t*

1 1 ( ) ( ) .......... 6 24 *t t*

( ) ( ) ( )( ) ( ) 12

Within the spectral region where the parameter *x* is real, the parametric gain is maximum

This means that the pump must be tuned at some frequency within the fiber anomalous

without amplification (Stolen & Bjorkholm, 1982). Here we change our notation and designate, the pump, the signal, and the idler powers as *P1*, *P2*, and *P*−,+ , and the angular

(Chavez Boggio et al., 2005a). In other words, the parametric gain will be high if

*L*

(

2

3 4

3 4

The diagram shown in Fig. 1 illustrates the principle of OAM-4F. Two co-polarized input signals at 1 and 2 are coupled into a fiber, where they co-propagate through a medium that favors the occurrence of FWM. When the fiber attenuation coefficient ≠ 0, Eq. (20) can be rewritten as:

$$P\_{-,+} = \eta \gamma^2 P\_{1,2}^2 P\_{2,1} \exp\left(-aL\right) \left[\frac{1 - \exp\left(-aL\right)}{a}\right]^2,\tag{21}$$

where *P*1, *P*2, *P*-, and *P*+ are the respective optical powers of the channels at frequencies 1, 2, − , and +, *L* is the fiber length, is the fiber nonlinear coefficient, and is the wavelength and intensity-dependent FWM generation efficiency, well described in the literature (Mussot et al., 2007), which is given by:

$$\eta = 4L^2 \left( \frac{\sin(\Delta \beta L)}{\Delta \beta L} \right)^2. \tag{22}$$

If the channels at *1* and *2* are codified with ideal on-off keying (OOK) modulation, then *P-,+* is null whenever one of these input channels transmits a 0-bit. Here, however, we assume that these channels are codified by a binary amplitude-shift keying (2-ASK) scheme, where the 0-bit powers of are intentionally offset. In this case, the extinction ratios (ER) corresponding to the channels at *1* and *2* are:

All-Optical Amplitude Multiplexing Through

whether 2 2 01 10

in (33). When 2 2 01 10

Similarly, if 2 2 01 10

12 12 *PP PP* 00 00 :

The analysis above can be repeated for the signal at

is photodetected by a circuit such as the one illustrated in Fig. 2.

Initially the signal is optically amplified and filtered at

analysis is left for the interested reader.

12 12 *PP PP* 00 00 :

one. On the other hand, 01

2 2 01 10

Fiber Parametric Interaction Between Binary Signals 9

*out P* depending, respectively, on

12 12 *PP PP* 00 00 .  In this case, Eqs. (24)- (27) would be

*+*; then, it is photo-detected by a

*out P* may be lower or higher than 10

12 12 *PP PP* 00 00 , the quaternary signal degenerates into a ternary one.

12 12 *PP PP* 00 00 or 2 2 01 10

The four-levels of quaternary signals give rise to an eye-diagram structure that comprises three eyes. We identify the eye made up of the two lowest power levels with the subscript "*low*"; analogously the subscripts "*int*" and "*up*" are utilized for the eyes that involve the two intermediate and the two higher power levels, respectively. For OAM-F4, it is possible to find the relative extinction ratios (RER) of such eyes, *rlow*, *rint*, and *rup*, by substituting (32)

01 00 2

10 01 2

11 10 2

10 00 2

01 10 2

11 01 2

Eqs. (24), (26) and (27) reveal some important properties of the generated 4-ASK signal. First, its powers do not depend on the phase of the input signals. Second, the power level distribution depends solely on the ERs of the two input signals. Finally, such power level distribution cannot be arbitrarily chosen. For instance, in the case where Eq. (26) hold, if one

modified, but the general properties of the generated 4-ASK signal would not change. Such

It is important to understand how information of the input binary signals may be recovered from the quaternary-amplitude one. To achieve this goal, it is assumed that the 4-ASK signal

PIN photo-diode with responsivity *RS*, low-pass filtered and submitted to an electronic

increases *r2*, both *rlow* and *rup* are enhanced; however, *rint* is simultaneously decreased.

Fig. 1. Scheme illustrating the principle of operation of OAM-4F.

$$r\_1 = P\_1^1(0) / P\_1^0(0) \tag{23a}$$

$$r\_2 = P\_2^1\left(0\right) / P\_2^0\left(0\right) \tag{23b}$$

where, 0 *j i P* designates the power of bit *j* (*j*= *0* or *1*) at the channel at *i* (*i*= *1* or *2*). Eq. (21) indicates that the signals filtered at *+* may assume four different power levels given by:

$$P\_{"out" }^{00} = k\_+ P\_1^0 \left( 0 \right) \left[ P\_2^0 \left( 0 \right) \right]^2 \tag{24a}$$

$$P\_{\text{out}+}^{01} = k\_+ P\_1^0(0) \left[ P\_2^1(0) \right]^2 \tag{24b}$$

$$P\_{\text{out}+}^{10} = k\_+ P\_1^1(0) \left[ P\_2^0(0) \right]^2 \tag{24c}$$

$$P\_{out+}^{11} = k\_+ P\_1^1(0) \left[ P\_2^1(0) \right]^2 \tag{24d}$$

where *out mn P* is the power of the signal envelope at *+* when the signal at *1* transmits a bit *m* (*m*= *0* or *1*) and the signal at *2* carries a bit *n* (*n*= *0* or *1*), and

$$k\_+ = \eta \gamma^2 \exp\left(-aL\right) \left[\frac{1 - \exp\left(-aL\right)}{a}\right]^2\tag{25}$$

Eqs. (24) clearly show that the signal formed at the fiber output, and selected by the optical band-pass filter (OBPF) centered at *+*, is a quaternary amplitude-shift keying (4-ASK) one. It should be noted that 00 *out P* is always the lowest power whereas 11 *out P* is always the highest

1 0

1 0

00 0 0 2

01 0 1 2

10 1 0 2

11 1 1 2

*2* carries a bit *n* (*n*= *0* or *1*), and

2 1 exp exp *L k L*

Eqs. (24) clearly show that the signal formed at the fiber output, and selected by the optical

*out P* is always the lowest power whereas 11

2

*i P* designates the power of bit *j* (*j*= *0* or *1*) at the channel at

*mn P* is the power of the signal envelope at

11 1 *rP P* 0/ 0 (23a)

22 2 *rP P* 0/ 0 (23b)

*+* may assume four different power levels given by:

1 2 0 0 *out P kP P* (24a)

1 2 0 0 *out P kP P* (24b)

1 2 0 0 *out P kP P* (24c)

1 2 0 0 *out P kP P* (24d)

*+*, is a quaternary amplitude-shift keying (4-ASK) one.

*+* when the signal at

(25)

*out P* is always the highest

*i* (*i*= *1* or *2*). Eq. (21)

*1* transmits a bit *m*

Fig. 1. Scheme illustrating the principle of operation of OAM-4F.

where, 0 *j*

where *out*

(*m*= *0* or *1*) and the signal at

It should be noted that 00

band-pass filter (OBPF) centered at

indicates that the signals filtered at

one. On the other hand, 01 *out P* may be lower or higher than 10 *out P* depending, respectively, on whether 2 2 01 10 12 12 *PP PP* 00 00 or 2 2 01 10 12 12 *PP PP* 00 00 . In case 2 2 01 10 12 12 *PP PP* 00 00 , the quaternary signal degenerates into a ternary one.

The four-levels of quaternary signals give rise to an eye-diagram structure that comprises three eyes. We identify the eye made up of the two lowest power levels with the subscript "*low*"; analogously the subscripts "*int*" and "*up*" are utilized for the eyes that involve the two intermediate and the two higher power levels, respectively. For OAM-F4, it is possible to find the relative extinction ratios (RER) of such eyes, *rlow*, *rint*, and *rup*, by substituting (32) in (33). When 2 2 01 10 12 12 *PP PP* 00 00 :

$$r\_{low} = P\_{out+}^{01} \int P\_{out+}^{00} = r\_2^2 \tag{26a}$$

$$r\_{\rm int} = P\_{\rm out}^{10} \int P\_{\rm out+}^{01} = r\_1 \Big/ r\_2^2 \tag{26b}$$

$$r\_{up} = P\_{\text{out}+}^{11} \int P\_{\text{out}+}^{10} = r\_2^2 \tag{26c}$$

Similarly, if 2 2 01 10 12 12 *PP PP* 00 00 :

$$r\_{low} = P\_{out+}^{10} \int P\_{out+}^{00} = r\_1^2 \tag{27a}$$

$$\mathbf{r}\_{\rm int} = \mathbf{P}\_{\rm out}^{01} \int \mathbf{P}\_{\rm out}^{10} = \mathbf{r}\_2 \Big/ \mathbf{r}\_1^2 \tag{27b}$$

$$r\_{\mu p} = P\_{\text{out}+}^{11} \int P\_{\text{out}+}^{01} = r\_1^2 \tag{27c}$$

Eqs. (24), (26) and (27) reveal some important properties of the generated 4-ASK signal. First, its powers do not depend on the phase of the input signals. Second, the power level distribution depends solely on the ERs of the two input signals. Finally, such power level distribution cannot be arbitrarily chosen. For instance, in the case where Eq. (26) hold, if one increases *r2*, both *rlow* and *rup* are enhanced; however, *rint* is simultaneously decreased.

The analysis above can be repeated for the signal at *-*. In this case, Eqs. (24)- (27) would be modified, but the general properties of the generated 4-ASK signal would not change. Such analysis is left for the interested reader.

It is important to understand how information of the input binary signals may be recovered from the quaternary-amplitude one. To achieve this goal, it is assumed that the 4-ASK signal is photodetected by a circuit such as the one illustrated in Fig. 2.

Initially the signal is optically amplified and filtered at *+*; then, it is photo-detected by a PIN photo-diode with responsivity *RS*, low-pass filtered and submitted to an electronic

All-Optical Amplitude Multiplexing Through

be estimated from the BER of the intermediate eye:

1 3

carriers at *f1*= 193.00 and *f2*= 193.15 THz, where *fi*= 2

**PC**

Fig. 3. Experimental setup for all-optical multiplexing.

1

**3.1.2 Results and discussion**

backscattering.

**PRBS1 TLS1**

**PRBS2 TLS2**

before being inputted to the DSA.

Fiber Parametric Interaction Between Binary Signals 11

*2* transmitted a bit 0 (1) whenever the two lower (upper) levels are detected. In this way, BER1 is now the average BER of the three eyes of the quaternary signal whereas BER2 may

>

*ii ii ii BER Q Q Q*

*i i BER Q*

We note that other reports suggest that the input binary signal may also be optically recovered from the 4-ASK signal with the use of transfer functions generated by self-phase modulation (Oda & Maruta, 2006) or FWM (Fagotto & Abbade, 2010) effects. However, a discussion concerning such all-optical approaches is beyond the scope of this chapter.

Fig. 3 illustrates the experimental setup used to perform amplitude multiplexing through FWM. Two 1 Gb/s 212-1 pseudorandom bit sequences (PRBS) directly modulate the optical

these signals are co-polarized and then amplified by an Erbium-doped fiber amplifier (EDFA), EDFA1, up to an average peak power of 12 dBm. In the sequence, they are launched into a dispersion-shifted fiber (DSF) with = 0.20 dB/km, 0= 1550 nm, dispersion slope S0= 0.074 ps/(nm.km), = 2.0 (W.km)-1, and L= 25.0 km. Since the powers at the fiber input are relatively low, it is not necessary to use any mechanism to prevent Brillouin

**DSF OBPF1**

Actually, since both signals are modulated, the Brillouin backscattering threshold should be a few dB higher than in the case of continuous-wave (cw) operation. Fiber FWM generates two sidebands, one at *f-*= 192.85 THz and other at *f+*= 193.30 THz. The latter is filtered by OBPF1, amplified by EDFA2, and then filtered again by OBPF2. Next, the signal is received by a digital signal analyzer (DSA). The double filtering is required because the first filter OBPF1 is not enough to eliminate effectively the input signals. Therefore, the OBPF1 output needs to be amplified and then filtered again by OBPF2

**G**

**EDFA1**

2

10 00 01 10 11 01

10 00 01 10 11 01

01 10

01 10

**G**

  (29a)

(29b)

*i* (*i*= 1, 2). Previous to being coupled,

**EDFA2 OBPF2 DSA**

Fig. 2. Multiamplitude signal detector.

decision circuit (EDC), whose purpose is to indicate which bits were transmitted by the signals at *1* and *2*. It is assumed that noise at EDC obeys a Gaussian distribution.

There are then two possibilities. The first one occurs when 2 2 01 10 12 12 *PP PP* 00 00

and it is illustrated in the left part of the inset of Fig. 1. A simple inspection of this figure indicates the following two detection rules should be utilized by the EDC: a) *1* transmitted a bit 0 (1) whenever the two lower (upper) levels are detected; and b) *2* transmitted a bit 0 (1) whenever the lower and third (second and fourth) power levels are detected.

These detection rules may be used to estimate the BERs of the binary signals extracted from the 4-ASK signals. To accomplish this goal, we first consider that the noise fluctuations between consecutive levels are much larger than the ones between non-consecutive levels. This hypothesis must hold for practical situations where even the noise fluctuations between adjacent levels must be low to keep the BERs at acceptable levels. Then, we observe from rule (a) and the left part of the inset of Fig. 1, that the BER for the signal at 1, *BER1*, is equivalent to the one of a binary signal with threshold level between the second and third levels. In this way:

$$BER\_1 = Q\left(\frac{i\_{10} - i\_{01}}{\sigma\_{10} + \sigma\_{01}}\right) \tag{28a}$$

where, 2 /2 1 2 *y x Q x e dy* is the complementary error function, *ixy*= *RS out xy P* is the average electronic current associated with power level *out xy P* , and *xy* is the *ixy* standard deviation. From rule (b) and the left part of the inset of Fig. 2, it is observed that the BER for the signal at *2*, *BER2*, is equivalent to the average BER of three binary signals with decision thresholds between the first (lower) and second; the second and third; and the third and fourth (highest) levels:

$$BER\_2 = \frac{1}{3} \left[ Q \left( \frac{i\_{01} - i\_{00}}{\sigma\_{01} + \sigma\_{00}} \right) + Q \left( \frac{i\_{10} - i\_{01}}{\sigma\_{10} + \sigma\_{01}} \right) + Q \left( \frac{i\_{11} - i\_{10}}{\sigma\_{11} + \sigma\_{10}} \right) \right] \tag{28b}$$

The second possibility occurs when 2 2 01 10 12 12 *PP PP* 00 00 and it is illustrated in the right part of the inset of Fig. 1. Following a procedure similar to the one described above and inspecting this figure, it is easy to verify that the detection rules are: c) *1* transmitted a bit 0 (1) whenever the lower and third (second and fourth) power levels are detected and d)

decision circuit (EDC), whose purpose is to indicate which bits were transmitted by the

There are then two possibilities. The first one occurs when 2 2 01 10

and it is illustrated in the left part of the inset of Fig. 1. A simple inspection of this figure

These detection rules may be used to estimate the BERs of the binary signals extracted from the 4-ASK signals. To accomplish this goal, we first consider that the noise fluctuations between consecutive levels are much larger than the ones between non-consecutive levels. This hypothesis must hold for practical situations where even the noise fluctuations between adjacent levels must be low to keep the BERs at acceptable levels. Then, we observe from

equivalent to the one of a binary signal with threshold level between the second and third

10 01

*e dy* is the complementary error function, *ixy*= *RS out*

deviation. From rule (b) and the left part of the inset of Fig. 2, it is observed that the BER for

thresholds between the first (lower) and second; the second and third; and the third and

the right part of the inset of Fig. 1. Following a procedure similar to the one described above

bit 0 (1) whenever the lower and third (second and fourth) power levels are detected and d)

*ii ii ii BERQ Q Q*

The second possibility occurs when 2 2 01 10

and inspecting this figure, it is easy to verify that the detection rules are: c)

10 01

*2*, *BER2*, is equivalent to the average BER of three binary signals with decision

01 00 10 01 11 10

01 00 10 01 11 10

*xy P*

12 12 *PP PP* 00 00 and it is illustrated in

, and

indicates the following two detection rules should be utilized by the EDC: a)

(1) whenever the lower and third (second and fourth) power levels are detected.

rule (a) and the left part of the inset of Fig. 1, that the BER for the signal at

1

*i i BER Q*

a bit 0 (1) whenever the two lower (upper) levels are detected; and b)

*2*. It is assumed that noise at EDC obeys a Gaussian distribution.

**LPF**

RS

Electronic Decision Circuit

12 12 *PP PP* 00 00

*1* transmitted

*2* transmitted a bit 0

*xy P*

(28b)

*1* transmitted a

*xy* is the *ixy* standard

is the

(28a)

1, *BER1*, is

**EDFA**

Fig. 2. Multiamplitude signal detector.

signals at

*1* and

levels. In this way:

the signal at

where, 2 /2 1 2 *y x Q x*

fourth (highest) levels:

2

1 3

average electronic current associated with power level *out*

**OBPF**

*2* transmitted a bit 0 (1) whenever the two lower (upper) levels are detected. In this way, BER1 is now the average BER of the three eyes of the quaternary signal whereas BER2 may be estimated from the BER of the intermediate eye:

$$BER\_1 = \frac{1}{3} \left[ Q \left( \frac{i\_{10} - i\_{00}}{\sigma\_{10} + \sigma\_{00}} \right) + Q \left( \frac{i\_{01} - i\_{10}}{\sigma\_{01} + \sigma\_{10}} \right) + Q \left( \frac{i\_{11} - i\_{01}}{\sigma\_{11} + \sigma\_{01}} \right) \right] \tag{29a}$$

$$BER\_2 = Q \left( \frac{i\_{01} - i\_{10}}{\sigma\_{01} + \sigma\_{10}} \right) \tag{29b}$$

We note that other reports suggest that the input binary signal may also be optically recovered from the 4-ASK signal with the use of transfer functions generated by self-phase modulation (Oda & Maruta, 2006) or FWM (Fagotto & Abbade, 2010) effects. However, a discussion concerning such all-optical approaches is beyond the scope of this chapter.

## **3.1.2 Results and discussion**

Fig. 3 illustrates the experimental setup used to perform amplitude multiplexing through FWM. Two 1 Gb/s 212-1 pseudorandom bit sequences (PRBS) directly modulate the optical carriers at *f1*= 193.00 and *f2*= 193.15 THz, where *fi*= 2*i* (*i*= 1, 2). Previous to being coupled, these signals are co-polarized and then amplified by an Erbium-doped fiber amplifier (EDFA), EDFA1, up to an average peak power of 12 dBm. In the sequence, they are launched into a dispersion-shifted fiber (DSF) with = 0.20 dB/km, 0= 1550 nm, dispersion slope S0= 0.074 ps/(nm.km), = 2.0 (W.km)-1, and L= 25.0 km. Since the powers at the fiber input are relatively low, it is not necessary to use any mechanism to prevent Brillouin backscattering.

Fig. 3. Experimental setup for all-optical multiplexing.

Actually, since both signals are modulated, the Brillouin backscattering threshold should be a few dB higher than in the case of continuous-wave (cw) operation. Fiber FWM generates two sidebands, one at *f-*= 192.85 THz and other at *f+*= 193.30 THz. The latter is filtered by OBPF1, amplified by EDFA2, and then filtered again by OBPF2. Next, the signal is received by a digital signal analyzer (DSA). The double filtering is required because the first filter OBPF1 is not enough to eliminate effectively the input signals. Therefore, the OBPF1 output needs to be amplified and then filtered again by OBPF2 before being inputted to the DSA.

All-Optical Amplitude Multiplexing Through

transmitted by the channel at

( 2 2 01 10

Fig 6(c) shows the eye diagrams for such situation.

constant and maximum value in agreement with (26).

Fiber Parametric Interaction Between Binary Signals 13

magnitude of power, represent levels 01 and 10 (where, as before, level *ij* stands for the bit *i*

is increased to 4.8 dB, *rint* in (26b) becomes close to unity and the two intermediate eyes get

very close; this is shown in Fig. 6b. If *r1* is further increased to 7.7 dB, then *r12*< *r2*

Fig. 6. Eye-diagrams for *r2*=2.6 dB and (a) *r1*=2.6 dB, (b) *r1*=4.8 dB, and (c) *r1*=7.7 dB.

12 12 *PP PP* 00 00 ) and the position between levels 01 and 10 is exchanged;

To complete such analysis, graphs of (a) *rup*, (b) *rint*, and (c) *rlow* as a function of *r1* are plotted in Fig. 7 for experimental data and theoretical curves, for *r2*= 2.6 dB. The agreement between such results is quite good. As predicted by (27), increasing *r1*, initially causes *rint* to decrease.

Fig. 7. Theoretical (line) and experimental (dots) ERs for the quaternary signal. When *r1*≈ *r2*2, *rint* achieves a minimum value; this is the point where the quaternary signal degenerates into a ternary-amplitude one and where levels 10 and 01 exchange their positions. As *r1* is further increased, (27) is no longer valid and (26) needs to be applied; in this region *rint* increases again. It is also observed that *rup*≈ *rlow* for any value of *r1*. As *r1* is increased, *rup* and *rlow* initially also increase, in agreement with (27). Then, for *r1≥ r2*2, *rup* and *rlow* remain at a

*1* and for the bit *j* transmitted by the channel at

*2*). When *r1*

The signal power spectra at (a) EDFA1 input, (b) DSF output, and (c) OBPF2 output are plotted in Fig. 4. An optical signal-to-noise ratio (OSNR) of 27 dB is achieved at the output of the second optical filter.

Fig. 4. Power Spectra.

Fig. 5 exhibits two unsynchronized PRBSs used to modulate the signals (a) at *f1* with *r1*= 4.0 dB and (b) at *f2* with *r2*= 1.7 dB, and the (c) quaternary signal obtained at *f+*. In this situation, *r1*> *r2*2 (which is equivalent to 2 2 01 10 12 12 *PP PP* 00 00 ) and the quaternary signal is governed by (26).

Fig. 5. (a) Binary input sequence at *f1* and (b) at *f2* and (c) quaternary output signal.

Fig. 6a shows the eye diagrams for *r2*= 2.6 dB, and *r1*= 2.6 dB. In this case, again *r1 2*> *r2* ( 2 2 01 10 12 12 *PP PP* 00 00 ) and so the two intermediate powers, in increasing

The signal power spectra at (a) EDFA1 input, (b) DSF output, and (c) OBPF2 output are plotted in Fig. 4. An optical signal-to-noise ratio (OSNR) of 27 dB is achieved at the output

**Wavelength (nm)**

12 12 *PP PP* 00 00 ) and the quaternary signal is

*2*> *r2*

Fig. 5 exhibits two unsynchronized PRBSs used to modulate the signals (a) at *f1* with *r1*= 4.0 dB and (b) at *f2* with *r2*= 1.7 dB, and the (c) quaternary signal obtained at *f+*. In this situation,

Fig. 5. (a) Binary input sequence at *f1* and (b) at *f2* and (c) quaternary output signal.

Fig. 6a shows the eye diagrams for *r2*= 2.6 dB, and *r1*= 2.6 dB. In this case, again *r1*

12 12 *PP PP* 00 00 ) and so the two intermediate powers, in increasing

*r1*> *r2*2 (which is equivalent to 2 2 01 10

of the second optical filter.

**Power(dBm)**

Fig. 4. Power Spectra.

governed by (26).

( 2 2 01 10

magnitude of power, represent levels 01 and 10 (where, as before, level *ij* stands for the bit *i* transmitted by the channel at *1* and for the bit *j* transmitted by the channel at *2*). When *r1* is increased to 4.8 dB, *rint* in (26b) becomes close to unity and the two intermediate eyes get very close; this is shown in Fig. 6b. If *r1* is further increased to 7.7 dB, then *r12*< *r2*

Fig. 6. Eye-diagrams for *r2*=2.6 dB and (a) *r1*=2.6 dB, (b) *r1*=4.8 dB, and (c) *r1*=7.7 dB.

( 2 2 01 10 12 12 *PP PP* 00 00 ) and the position between levels 01 and 10 is exchanged; Fig 6(c) shows the eye diagrams for such situation.

To complete such analysis, graphs of (a) *rup*, (b) *rint*, and (c) *rlow* as a function of *r1* are plotted in Fig. 7 for experimental data and theoretical curves, for *r2*= 2.6 dB. The agreement between such results is quite good. As predicted by (27), increasing *r1*, initially causes *rint* to decrease.

Fig. 7. Theoretical (line) and experimental (dots) ERs for the quaternary signal. When *r1*≈ *r2*2, *rint* achieves a minimum value; this is the point where the quaternary signal degenerates into a ternary-amplitude one and where levels 10 and 01 exchange their positions. As *r1* is further increased, (27) is no longer valid and (26) needs to be applied; in this region *rint* increases again. It is also observed that *rup*≈ *rlow* for any value of *r1*. As *r1* is increased, *rup* and *rlow* initially also increase, in agreement with (27). Then, for *r1≥ r2*2, *rup* and *rlow* remain at a constant and maximum value in agreement with (26).

All-Optical Amplitude Multiplexing Through

Fig. 8. Scheme for (a) PA and (b) ASK-4 generation. (

dispersion wavelength of the fiber).

input are designated as 0(0) *PP* and 1

For example, consider that the signal at

considerably higher than 1

frequencies of the pump, the signal and the idler, and

but two new features are added to the signals involved.

"1" bit. Consequently, each of the binary levels of the signal at

higher gain, splitting into two new power levels at the fiber output.

Fiber Parametric Interaction Between Binary Signals 15

scheme of the theoretical principle of OAM-4P. Essentially, this principle is equal to the PA,

First, as in OAM-4, the power of the probe signal is intentionally offset from zero. Thus, the ER of the signal is no longer infinite (in contrast to the ideal OOK signals), and is given by:

Second, the pump is also modulated with binary information, in an analogous way to that of the probe signal. The powers of the "0" and "1" bits transmitted by the pump at the fiber

amplification factors, 0 *GP* or 1 *GP* , depending on whether the pump information is a "0" or

modulation will split it in two power levels "01" and "11", depending on whether the pump information is a bit "0" or "1", respectively. This argument is also valid when the probe

*k* with *k* = *P*, *S*, *i* represents the angular

0 = 2c/*λ0*, where 0 is the zero

1 0 (0) / (0) *SS S rP P* . (30)

(0) *PP* , respectively. It is assumed that 0(0) *PP* is

*S* has the "1" bit power level, then the pump

*S* will have a lower or a

(0) *PS* . Once the pump is modulated, it will provide two

It should be noted that the FWM equations presented in Section 2 assume that the involved signals are in *cw* regime. The accordance between theoretical results and experimental data observed in this subsection suggest that such FWM equations are also valid when the involved signals are modulated.

We reckon that the first experimental results concerning this technique were reported in (Xu et al., 2005). There, two 10 Gb/s signals were multiplexed in a 20 Gb/s 4-ASK one, which was then transmitted through a 160 km single mode fiber (SMF) link; after such propagation distance, chromatic dispersion was compensated. The eye-penalties experienced by the quaternary signal were relatively small. In another experiment (Abbade et al, 2006), a 2 Gb/s 4-ASK was generated and propagated through a 20-km long SMF link in a field-trial network (FAPESP, n.d.). Eqs. (28) and (29) were utilized to estimate the BERs for signals with different values of *r1* and *r2*. Although chromatic dispersion was not compensated and no special care concerning the bandwidth of filters was taken into account, *BER1*< 10-9 and *BER2*< 10-12 were obtained for several combinations of values of *r1* and *r2*. This indicates that the 4-ASK signal generated by FWM amplitude multiplexing could be properly utilized in practical applications.

BER performance of multi-amplitude signals depends on the dominant kind of noise. Generally speaking, different types of noise require different power level distributions to minimize signal BER. Theoretical and experimental analyses of such optimum distribution for 4-ASK signals are presented in (Walklin & Conradi, 1999). In particular, it is shown that under the dominance of amplified spontaneous emission (ASE) noise, the power levels of a quaternary amplitude signal should be in proportions of 0: 1: 4: 9.

Concerning the 4-ASK generated by OAM-4F, such optimal distribution would apply to the cases described by Eqs. (28b) and (29a), where consecutive power levels represent complementary bits (*i.e.*, 0101). Unfortunately, the proportions of 0: 1: 4: 9 cannot be achieved by FWM amplitude multiplexing. This may be understood by noting in Eq. (24a) that 00 *out P* is never null; moreover, *rup*= *rlow* ((26) and (27)). In the cases described by Eqs. (28a) and (29b), the two lowest power levels stand for bit 0, whereas the two highest ones represent a bit 1. Therefore, for a given average power, BER minimization would require the two lowest power levels and also the two highest power levels to coincide. But this degenerates the 4-ASK signal into a 2-ASK one, which does not allow information from the signals at *1* and *2* to be recovered. This discussion shows that minimizing *BER1* does not lead to a minimization of *BER2*.

## **3.2 Optical amplitude multiplexing through fiber parametric amplification**

#### **3.2.1 Theory**

As detailed in Section 2, fiber optics parametric amplifiers are realized by coupling both a weak probe signal at angular frequency *S*, and a high-power pump signal at angular frequency *P* into an optical fiber that acts as a nonlinear medium. Usually the performance of parametric amplifiers is analyzed for binary signals with two intensity levels representing bits "0" and "1" (Kalogerakis et al., 2006; Mussot et al., 2007; Chavez Boggio et al., 2004b; Peucheret et al., 2006) as sketched in Fig. 8(a). The power of these levels at the fiber input are indicated by 0(0) *PS* and 1 (0) *PS* , respectively; the ideal 0(0) *PS* being null. Figure 8(b) shows a

It should be noted that the FWM equations presented in Section 2 assume that the involved signals are in *cw* regime. The accordance between theoretical results and experimental data observed in this subsection suggest that such FWM equations are also valid when the

We reckon that the first experimental results concerning this technique were reported in (Xu et al., 2005). There, two 10 Gb/s signals were multiplexed in a 20 Gb/s 4-ASK one, which was then transmitted through a 160 km single mode fiber (SMF) link; after such propagation distance, chromatic dispersion was compensated. The eye-penalties experienced by the quaternary signal were relatively small. In another experiment (Abbade et al, 2006), a 2 Gb/s 4-ASK was generated and propagated through a 20-km long SMF link in a field-trial network (FAPESP, n.d.). Eqs. (28) and (29) were utilized to estimate the BERs for signals with different values of *r1* and *r2*. Although chromatic dispersion was not compensated and no special care concerning the bandwidth of filters was taken into account, *BER1*< 10-9 and *BER2*< 10-12 were obtained for several combinations of values of *r1* and *r2*. This indicates that the 4-ASK signal generated by FWM amplitude multiplexing could be properly utilized in

BER performance of multi-amplitude signals depends on the dominant kind of noise. Generally speaking, different types of noise require different power level distributions to minimize signal BER. Theoretical and experimental analyses of such optimum distribution for 4-ASK signals are presented in (Walklin & Conradi, 1999). In particular, it is shown that under the dominance of amplified spontaneous emission (ASE) noise, the power levels of a

Concerning the 4-ASK generated by OAM-4F, such optimal distribution would apply to the cases described by Eqs. (28b) and (29a), where consecutive power levels represent complementary bits (*i.e.*, 0101). Unfortunately, the proportions of 0: 1: 4: 9 cannot be achieved by FWM amplitude multiplexing. This may be understood by noting in Eq. (24a)

As detailed in Section 2, fiber optics parametric amplifiers are realized by coupling both a

of parametric amplifiers is analyzed for binary signals with two intensity levels representing bits "0" and "1" (Kalogerakis et al., 2006; Mussot et al., 2007; Chavez Boggio et al., 2004b; Peucheret et al., 2006) as sketched in Fig. 8(a). The power of these levels at the fiber input are

*P* into an optical fiber that acts as a nonlinear medium. Usually the performance

(0) *PS* , respectively; the ideal 0(0) *PS* being null. Figure 8(b) shows a

**3.2 Optical amplitude multiplexing through fiber parametric amplification**

*out P* is never null; moreover, *rup*= *rlow* ((26) and (27)). In the cases described by Eqs. (28a) and (29b), the two lowest power levels stand for bit 0, whereas the two highest ones represent a bit 1. Therefore, for a given average power, BER minimization would require the two lowest power levels and also the two highest power levels to coincide. But this degenerates the 4-ASK signal into a 2-ASK one, which does not allow information from the

*2* to be recovered. This discussion shows that minimizing *BER1* does not

*S*, and a high-power pump signal at angular

quaternary amplitude signal should be in proportions of 0: 1: 4: 9.

involved signals are modulated.

practical applications.

that 00

signals at

**3.2.1 Theory**

frequency

*1* and

indicated by 0(0) *PS* and 1

lead to a minimization of *BER2*.

weak probe signal at angular frequency

Fig. 8. Scheme for (a) PA and (b) ASK-4 generation. (*k* with *k* = *P*, *S*, *i* represents the angular frequencies of the pump, the signal and the idler, and 0 = 2c/*λ0*, where 0 is the zero dispersion wavelength of the fiber).

scheme of the theoretical principle of OAM-4P. Essentially, this principle is equal to the PA, but two new features are added to the signals involved.

First, as in OAM-4, the power of the probe signal is intentionally offset from zero. Thus, the ER of the signal is no longer infinite (in contrast to the ideal OOK signals), and is given by:

$$r\_S = P\_S^1(0) / P\_S^0(0) \,. \tag{30}$$

Second, the pump is also modulated with binary information, in an analogous way to that of the probe signal. The powers of the "0" and "1" bits transmitted by the pump at the fiber input are designated as 0(0) *PP* and 1 (0) *PP* , respectively. It is assumed that 0(0) *PP* is considerably higher than 1 (0) *PS* . Once the pump is modulated, it will provide two amplification factors, 0 *GP* or 1 *GP* , depending on whether the pump information is a "0" or "1" bit. Consequently, each of the binary levels of the signal at *S* will have a lower or a higher gain, splitting into two new power levels at the fiber output.

For example, consider that the signal at *S* has the "1" bit power level, then the pump modulation will split it in two power levels "01" and "11", depending on whether the pump information is a bit "0" or "1", respectively. This argument is also valid when the probe

All-Optical Amplitude Multiplexing Through

and 1 *G P r G* .

and

not depend on the powers of the probe signal at

experiments, showing that our assumptions work fine.

**3.2.2 Results and discussions**

*k* = 2*c*/

**PRBS PM EDFA1**

**Pump**

Fig. 9. Experimental setup.

*f1 f2 f3*

**PC PC**

*S* = 1552.2 nm (

**Probe signal**

*rS* and *rG* it is possible to regulate the power of the quaternary levels.

Fiber Parametric Interaction Between Binary Signals 17

01 int 10

11 01 *out up G out P r r P*

Eqs. (32) and (34) show that the power level distribution of the generated 4-ASK signal does

utilized in the pump. This is quite reasonable because in our analytical model the power of the probe signal was supposed to be much lower than the pump power, and so it was neglected (Song et al., 1999a). Besides that, it is clear that by controlling only the parameters

The pump ER 1 0 (0)/ (0) *PP P rP P* is controlled by setting the values of the pump average powers, and then it is possible to control the value of *rG*. For the ideal OOK modulation in the pump ( 0(0) 0 *PP* ), Eqs. (31), (32) and (34) are still valid, and in this special case 0 1 *GP*

The theoretical model for 4-ASK presented here and applied to modulated signals is actually based on a set of equations generated for *cw* signals. In the next two subsections the applicability of the technique to practical situations is tested through simulations and

Fig. 9 shows the experimental setup used in both simulations and experiments. The lasers

k, *k* = *P* or *S*), respectively.

**OBPF1**

A pseudo-random bit sequence at 1 Gb/s was used to modulate the lasers through direct modulation. In order to suppress stimulated Brillouin scattering (SBS), the pump linewidth was broadened by phase modulation using a phase modulator (PM) driven by three RF signals. After the Erbium-doped fiber amplifier (EDFA 1), the pump was filtered with an

**DS fiber 90 20**

**10 80**

(two DFB lasers) that were used as the pump and the signal were tuned at

**PRBS EDFA2**

**PC**

*r*

*out S out G P r*

*P r* (34b)

(34c)

*S* (only on its ER), but on the powers

**OBPF2 OBPF3 DSA**

**Optical Spectrum Analyser**

*P* = 1553.5 nm

signal transports a bit "0" power level. In such case, the probe signal will either assume a "00" or "10" power level at the fiber output, depending again on whether the pump information is a bit "0" or "1", respectively. Therefore, the power levels of the signal envelope at *S* and at *z* = *L* will be:

$$P\_{out}^{00} = G\_P^0 P\_S^0 \text{(0)}\tag{31a}$$

$$P\_{out}^{01} = \mathbb{G}\_P^0 P\_S^1(\mathbf{0})\tag{31b}$$

$$P\_{out}^{10} = G\_P^1 P\_S^0 \text{(0)}\tag{31c}$$

$$P\_{out}^{11} = G\_P^1 P\_S^1 \text{(0)}\tag{31d}$$

where *ij Pout* represents the power of the signal at *S* and at *z* = *L*, when bits *i* and *j* (*i*, *j* = "0" or "1") at *z* = 0 are assigned to the pump and the probe signal, respectively. 0 *GP* and 1 *GP* can be obtained through Eq. (16) with pump powers 0 *PP* and 1 *PP* , respectively.

Note that Eqs. (31) clearly show that the 4-ASK signal contains information from both the pump and the probe signal. As in Section 3.1, we identify the eye made up of the two lowest power levels with the subscript "*low*"; analogously, the subscripts "*int*" and "*up*" are utilized for the eyes that involve the two intermediate and the two higher power levels, respectively. By using this convention and considering 10 01 *P P out out* ( 01 10 (0) (0) *GP GP PS PS* ), the ERs for each one of these eyes can be written as:

$$r\_{low} = \frac{P\_{out}^{01}}{P\_{out}^{00}} = r\_S \tag{32a}$$

$$r\_{\rm int} = \frac{P\_{out}^{10}}{P\_{out}^{01}} = \frac{r\_G}{r\_S} \tag{32b}$$

$$r\_{up} = \frac{P\_{out}^{11}}{P\_{out}^{10}} = r\_{\rm S} \tag{32c}$$

where *rG* is defined as

$$r\_{\rm G} = \frac{G\_P^1}{G\_P^0} \,\prime \tag{33}$$

and it is the ER associated with the amplification factors 0 *GP* and 1 *GP* . In the case 10 01 *P P out out* ( 01 1 0 *GP GP PS PS* (0) (0) ), the ERs are given by:

$$
\eta\_{low} = \frac{P\_{out}^{10}}{P\_{out}^{00}} = \eta\_{\text{G}} \tag{34a}
$$

signal transports a bit "0" power level. In such case, the probe signal will either assume a "00" or "10" power level at the fiber output, depending again on whether the pump information is a bit "0" or "1", respectively. Therefore, the power levels of the signal

or "1") at *z* = 0 are assigned to the pump and the probe signal, respectively. 0 *GP* and 1 *GP*

Note that Eqs. (31) clearly show that the 4-ASK signal contains information from both the pump and the probe signal. As in Section 3.1, we identify the eye made up of the two lowest power levels with the subscript "*low*"; analogously, the subscripts "*int*" and "*up*" are utilized for the eyes that involve the two intermediate and the two higher power levels, respectively. By using this convention and considering 10 01 *P P out out* ( 01 10 (0) (0) *GP GP PS PS* ), the

> 01 00 *out low S out P r r P*

10 int 01 *out G out S*

11 10 *out up S out P r r P*

> 1 0 *P*

*P G*

*G*

10 00 *out low G out P r r P*

and it is the ER associated with the amplification factors 0 *GP* and 1 *GP* . In the case 10 01 *P P out out*

*G*

*r*

can be obtained through Eq. (16) with pump powers 0 *PP* and 1 *PP* , respectively.

00 0 0(0) *P GP out P S* (31a)

01 0 1(0) *P GP out P S* (31b)

10 1 0(0) *P GP out P S* (31c)

11 1 1(0) *P GP out P S* (31d)

(32a)

*P r r P r* (32b)

(32c)

, (33)

(34a)

*S* and at *z* = *L*, when bits *i* and *j* (*i*, *j* = "0"

envelope at

*S* and at *z* = *L* will be:

where *ij Pout* represents the power of the signal at

ERs for each one of these eyes can be written as:

( 01 1 0 *GP GP PS PS* (0) (0) ), the ERs are given by:

where *rG* is defined as

$$r\_{\rm int} = \frac{P\_{out}^{01}}{P\_{out}^{10}} = \frac{r\_S}{r\_G} \tag{34b}$$

$$r\_{\text{top}} = \frac{P\_{out}^{11}}{P\_{out}^{01}} = r\_{\text{G}} \tag{34c}$$

Eqs. (32) and (34) show that the power level distribution of the generated 4-ASK signal does not depend on the powers of the probe signal at *S* (only on its ER), but on the powers utilized in the pump. This is quite reasonable because in our analytical model the power of the probe signal was supposed to be much lower than the pump power, and so it was neglected (Song et al., 1999a). Besides that, it is clear that by controlling only the parameters *rS* and *rG* it is possible to regulate the power of the quaternary levels.

The pump ER 1 0 (0)/ (0) *PP P rP P* is controlled by setting the values of the pump average powers, and then it is possible to control the value of *rG*. For the ideal OOK modulation in the pump ( 0(0) 0 *PP* ), Eqs. (31), (32) and (34) are still valid, and in this special case 0 1 *GP* and 1 *G P r G* .

The theoretical model for 4-ASK presented here and applied to modulated signals is actually based on a set of equations generated for *cw* signals. In the next two subsections the applicability of the technique to practical situations is tested through simulations and experiments, showing that our assumptions work fine.

## **3.2.2 Results and discussions**

Fig. 9 shows the experimental setup used in both simulations and experiments. The lasers (two DFB lasers) that were used as the pump and the signal were tuned at *P* = 1553.5 nm and *S* = 1552.2 nm (*k* = 2*c*/k, *k* = *P* or *S*), respectively.

Fig. 9. Experimental setup.

A pseudo-random bit sequence at 1 Gb/s was used to modulate the lasers through direct modulation. In order to suppress stimulated Brillouin scattering (SBS), the pump linewidth was broadened by phase modulation using a phase modulator (PM) driven by three RF signals. After the Erbium-doped fiber amplifier (EDFA 1), the pump was filtered with an

All-Optical Amplitude Multiplexing Through

performance of the quaternary signal is expected.

signals obtained in the experimental results.

good agreement for *rup* and *rint*.

at

Fiber Parametric Interaction Between Binary Signals 19

Fig. 10. (a) Experimental and simulated optical spectrum at the fiber output. (b) Binary pump. (c) Binary signal before amplification. (d) ASK-4 signal obtained after PA

gain; however, the gain bandwidth does not remain the same when the pump transmits either a bit "1" or "0", thus the change in bandwidth also modifies the spectral regions where the PM affects the signal performance. Consequently, some influence on the OSNR

Two bit sequences carried by the pump and the probe signal at the fiber input are illustrated in Figs. 10(b) and (c). The resulting quaternary signal is illustrated in Fig. 10(d). The pump and the probe signal ERs are *rP* = 6.0 dB and *rS* = 2.6 dB, respectively. As expected the signal of Fig. 10(d) presents four well-defined power levels. Note that the same power levels are obtained when the bits of the pump and the probe signals are repeated, and in the experimental case, the probe signal is slightly delayed in relation to the pump, which does not occur in the simulated case. This fact explains the slight delay between the quaternary

Fig. 11 shows the values of *rup*, *rint* and *rlow* as a function of *rS*, for *rG* varying from 3 to 7 dB. These values were calculated using the sets of Eqs. (32) and (34), and the analytical, the simulated, and the experimental values were plotted in all of these cases, showing a rather

However, in the case of *rlow*, a difference between the experimental and the analytical/simulated values for *rS*> 6 dB appeared. Such difference increases when the value of *rG* increases. Moreover, the experimental values are always smaller than those of the simulated ones and the difference is always smaller than 1.4 dB. This difference can be explained by the fact that when *rG* is significantly high, the quaternary signal ER, given by *low up G S* int *r r r r rr* is also high. Thus, even when the measurements are taken with a fixed average power, the power of level "00" can reach a value around -10 dBm; this value is comparable to the power level generated by the DSA photodiode dark current. Consequently, measurements of *rlow* are not as accurate as those for smaller values of *rG*.

To measure the *rS* values, the pump was simply turned off. Then, the ER of the binary signal

modulated pump with ER = *rP* were used to obtain *rG*. The PA gave two gains to the *cw*

*S* and a

*S* was measured from the eye-diagram at the DSA. A continuous-wave (*cw*) at

optical band-pass filter (OBPF) to suppress most of the amplified spontaneous emission (ASE). A 90/10 coupler was used to couple the pump and the signal to a 7 km long segment of dispersion shifted fiber (DSF).

The other fiber parameters were: the zero-dispersion wavelength *0* = 1550.2 nm, the zerodispersion wavelength variation Δ*0*= 0.08 nm, the dispersion slope *S0* = 0.074 ps/nm2/km, the nonlinear coefficient = 2.1 W-1km-1, and the attenuation coefficient = 0.2 dB/km. The maximization of the parametric gain was achieved by aligning the pump and the signal states of polarization using polarization controllers (PCs). The optical power was divided with an optical coupler at the fiber output for spectral and systemic characterization. A fraction of ~20% was delivered to an optical spectrum analyzer (OSA) for recording the spectra. The remaining power (~80%) was sent to an OBPF centered at the signal frequency *S*. The signal was then amplified and filtered again to reduce the ASE power accumulated in the amplification stage. Finally, the signal with an average power of approximately 0 dBm entered a digital oscilloscope to characterize the resulting ASK-4 eye-diagrams. The quaternary signal was also analyzed after being transmitted through a 75 km-long spool of standard fiber. To compensate for the dispersion due to the propagation, the signal was also passed through a -68 ps/nm compensating fiber.

An accurate control of the ERs of both the pump and the signal was a necessary item for the proper evaluation of the proposed technique. Thus, with the equipment available in our labs, we found that the required accuracy was more easily obtained by programming the modulation index of directly modulated lasers than by varying the bias voltage of the external modulator. However, the technique should also hold for signals with external modulation if specific modulators were available. It is anticipated that due to the femto-second response of PA (Grudinin et al., 1987) the technique should also work well for higher bit rates.

The simulated results presented here were obtained using exactly the same setup as used for the experiments. To perform the simulations as close as possible to the experiments, the laser linewidth was set at the maximum value of the equipment at ~30 MHz. Also the same phase modulation scheme used experimentally was added to our simulations. This point is particularly important because the PM induces noise at the level "1" of the binary eyediagrams. Consequently, additional noise is also expected at quaternary levels. The DSF was divided into 10 segments to perform the simulations in order to consider the influence of the zero dispersion wavelength variations along the fiber. All segments had the same length and the Δ*0* was distributed within the range *0*-Δ*0*/2 to *0*+ Δ*0*/2. The segments were randomly ordered. The propagation along each segment was handled by solving the nonlinear Schrödinger equation using the split-step Fourier method (Agrawal, 2001). A commercially available software program was used to perform the simulations.

Fig. 10 shows spectral and temporal results. Fig. 10(a) shows a comparison between the simulated and the experimental spectra at the fiber output. Note that the main difference between them is in the noise region of the spectra. The principal reason for such a difference is that the ASE originated in the pump Erbium booster amplifier, that is not completely suppressed by the band-pass filter, is then amplified by the parametric amplifier. It is important to note that *S* is placed within the frequency region where the signal performance is not affected by the phase modulation used to suppress the SBS (Boggio et al., 2005a). This region for conventional (non-modulated pump) PA is the region of maximum

optical band-pass filter (OBPF) to suppress most of the amplified spontaneous emission (ASE). A 90/10 coupler was used to couple the pump and the signal to a 7 km long segment

maximization of the parametric gain was achieved by aligning the pump and the signal states of polarization using polarization controllers (PCs). The optical power was divided with an optical coupler at the fiber output for spectral and systemic characterization. A fraction of ~20% was delivered to an optical spectrum analyzer (OSA) for recording the spectra. The remaining power (~80%) was sent to an OBPF centered at the signal frequency

*S*. The signal was then amplified and filtered again to reduce the ASE power accumulated in the amplification stage. Finally, the signal with an average power of approximately 0 dBm entered a digital oscilloscope to characterize the resulting ASK-4 eye-diagrams. The quaternary signal was also analyzed after being transmitted through a 75 km-long spool of standard fiber. To compensate for the dispersion due to the propagation, the signal was also

An accurate control of the ERs of both the pump and the signal was a necessary item for the proper evaluation of the proposed technique. Thus, with the equipment available in our labs, we found that the required accuracy was more easily obtained by programming the modulation index of directly modulated lasers than by varying the bias voltage of the external modulator. However, the technique should also hold for signals with external modulation if specific modulators were available. It is anticipated that due to the femto-second response of

The simulated results presented here were obtained using exactly the same setup as used for the experiments. To perform the simulations as close as possible to the experiments, the laser linewidth was set at the maximum value of the equipment at ~30 MHz. Also the same phase modulation scheme used experimentally was added to our simulations. This point is particularly important because the PM induces noise at the level "1" of the binary eyediagrams. Consequently, additional noise is also expected at quaternary levels. The DSF was divided into 10 segments to perform the simulations in order to consider the influence of the zero dispersion wavelength variations along the fiber. All segments had the same length

> *0*-Δ*0*/2 to *0*+ Δ

randomly ordered. The propagation along each segment was handled by solving the nonlinear Schrödinger equation using the split-step Fourier method (Agrawal, 2001). A

Fig. 10 shows spectral and temporal results. Fig. 10(a) shows a comparison between the simulated and the experimental spectra at the fiber output. Note that the main difference between them is in the noise region of the spectra. The principal reason for such a difference is that the ASE originated in the pump Erbium booster amplifier, that is not completely suppressed by the band-pass filter, is then amplified by the parametric amplifier. It is

performance is not affected by the phase modulation used to suppress the SBS (Boggio et al., 2005a). This region for conventional (non-modulated pump) PA is the region of maximum

*S* is placed within the frequency region where the signal

PA (Grudinin et al., 1987) the technique should also work well for higher bit rates.

commercially available software program was used to perform the simulations.

*0*= 0.08 nm, the dispersion slope *S0* = 0.074 ps/nm2/km,

*0* = 1550.2 nm, the zero-

*0*/2. The segments were

= 0.2 dB/km. The

The other fiber parameters were: the zero-dispersion wavelength

the nonlinear coefficient = 2.1 W-1km-1, and the attenuation coefficient

of dispersion shifted fiber (DSF).

dispersion wavelength variation Δ

passed through a -68 ps/nm compensating fiber.

*0* was distributed within the range

and the Δ

important to note that

Fig. 10. (a) Experimental and simulated optical spectrum at the fiber output. (b) Binary pump. (c) Binary signal before amplification. (d) ASK-4 signal obtained after PA

gain; however, the gain bandwidth does not remain the same when the pump transmits either a bit "1" or "0", thus the change in bandwidth also modifies the spectral regions where the PM affects the signal performance. Consequently, some influence on the OSNR performance of the quaternary signal is expected.

Two bit sequences carried by the pump and the probe signal at the fiber input are illustrated in Figs. 10(b) and (c). The resulting quaternary signal is illustrated in Fig. 10(d). The pump and the probe signal ERs are *rP* = 6.0 dB and *rS* = 2.6 dB, respectively. As expected the signal of Fig. 10(d) presents four well-defined power levels. Note that the same power levels are obtained when the bits of the pump and the probe signals are repeated, and in the experimental case, the probe signal is slightly delayed in relation to the pump, which does not occur in the simulated case. This fact explains the slight delay between the quaternary signals obtained in the experimental results.

Fig. 11 shows the values of *rup*, *rint* and *rlow* as a function of *rS*, for *rG* varying from 3 to 7 dB. These values were calculated using the sets of Eqs. (32) and (34), and the analytical, the simulated, and the experimental values were plotted in all of these cases, showing a rather good agreement for *rup* and *rint*.

However, in the case of *rlow*, a difference between the experimental and the analytical/simulated values for *rS*> 6 dB appeared. Such difference increases when the value of *rG* increases. Moreover, the experimental values are always smaller than those of the simulated ones and the difference is always smaller than 1.4 dB. This difference can be explained by the fact that when *rG* is significantly high, the quaternary signal ER, given by *low up G S* int *r r r r rr* is also high. Thus, even when the measurements are taken with a fixed

average power, the power of level "00" can reach a value around -10 dBm; this value is comparable to the power level generated by the DSA photodiode dark current. Consequently, measurements of *rlow* are not as accurate as those for smaller values of *rG*.

To measure the *rS* values, the pump was simply turned off. Then, the ER of the binary signal at *S* was measured from the eye-diagram at the DSA. A continuous-wave (*cw*) at *S* and a modulated pump with ER = *rP* were used to obtain *rG*. The PA gave two gains to the *cw*

All-Optical Amplitude Multiplexing Through

Fiber Parametric Interaction Between Binary Signals 21

Fig. 12. Simulated (full lines) and experimental (white circles) eye diagrams for (a) *r*S = 2.2

Fig. 13. Simulated (full lines) and experimental (white circles) eye diagrams for (a) *rS* = 2.2

A very good agreement between the simulated and experimental eye diagrams is observed in Figs. 12 and 13 for the two values of *rG* used, and similar results were observed for other

The eye diagrams for the quaternary signal that propagated through 75 km of standard fiber are shown in Fig. 14. The agreement between experimental eye diagrams and numerical results is also very good. The experimental BERs are lower than 10-13 and 10-11 before and after transmission, respectively. In principle, it is possible to improve such values by using a narrower optical band-pass filter or by increasing the quaternary signal average power. Table 1 presents estimations for the BERs before and after propagation by

Before Propagation After Propagation

Experiment Simulation Experiment Simulation

1.1x10-14 1.7x10-14 7.4x10-12 3.2x10-12

8.0x10-14 4.8x10-14 7.6x10-12 2.8x10-11

dB, (b) *r*S = 3.8 dB, and (c) *r*S = 6.9 dB. In all the cases *r*G = 4 dB.

dB, (b) *rS* = 5.1 dB, and (c) *rS* = 6.9 dB. In all the cases *rG* = 5 dB.

Table 1. BER estimation before and after fiber propagation.

values of *rG*.

using Eqs. (29).

*P BER*

*S BER*

Fig. 11. Values of *rup*, *rint* and *r*low as a function of *r*S for: (a) *r*G = 3.0 dB, (b) *r*G = 5.0 dB, (c) rG = 7.0 dB.

signal, generating a binary signal whose eye diagram can be seen on the DSA. The ER of such signal is precisely *rG*.

Two sets of three ASK-4 eye-diagrams for the signal at frequency *S* are exhibited in Figures 12 and 13. The labels "00" and "11" used to identify the quaternary pattern levels always correspond to the lowest and uppermost levels, respectively. On the other hand, the labels "01" or "10" are always used to identify the two intermediate levels, but in this case the label of each specific level depends on the value of the signal extinction ratio *rS*. For instance, for *rS* = 2.2 dB (as in Fig. 12(a) the second and third power levels of the quaternary eye diagram correspond to labels "01" and "10", respectively. This can be explained as follows. When *rS* = 2.2 dB and the pump level is "1", the parametric gain obtained by the signal level "0" is higher than that of the signal level "1" when the pump level is "0".

Fig. 12(b) shows a particular case when the parametric gain given by the pump level "1" to the signal level "0" coincides with that given by the pump level "0" to probe level "1". In such a situation, *rS rG* (3.8 dB 4 dB), and the quaternary signal degenerates into a ternary one. This situation must be avoided in practical applications. Fig. 12(c) presents the case *rS* = 6.8 dB > *rG*, which is the inverse case of the one shown in Fig. 12(a). Now, the power level of the label "10" is lower than that of the label "01". This is because the gain of pump level "1" given to level "0" is higher than that given by the pump level "0" to signal level "1".

Fig. 11. Values of *rup*, *rint* and *r*low as a function of *r*S for: (a) *r*G = 3.0 dB, (b) *r*G = 5.0 dB, (c) rG =

signal, generating a binary signal whose eye diagram can be seen on the DSA. The ER of

12 and 13. The labels "00" and "11" used to identify the quaternary pattern levels always correspond to the lowest and uppermost levels, respectively. On the other hand, the labels "01" or "10" are always used to identify the two intermediate levels, but in this case the label of each specific level depends on the value of the signal extinction ratio *rS*. For instance, for *rS* = 2.2 dB (as in Fig. 12(a) the second and third power levels of the quaternary eye diagram correspond to labels "01" and "10", respectively. This can be explained as follows. When *rS* = 2.2 dB and the pump level is "1", the parametric gain obtained by the signal level

Fig. 12(b) shows a particular case when the parametric gain given by the pump level "1" to the signal level "0" coincides with that given by the pump level "0" to probe level "1". In such a situation, *rS rG* (3.8 dB 4 dB), and the quaternary signal degenerates into a ternary one. This situation must be avoided in practical applications. Fig. 12(c) presents the case *rS* = 6.8 dB > *rG*, which is the inverse case of the one shown in Fig. 12(a). Now, the power level of the label "10" is lower than that of the label "01". This is because the gain of pump level "1"

given to level "0" is higher than that given by the pump level "0" to signal level "1".

*S* are exhibited in Figures

Two sets of three ASK-4 eye-diagrams for the signal at frequency

"0" is higher than that of the signal level "1" when the pump level is "0".

7.0 dB.

such signal is precisely *rG*.

Fig. 12. Simulated (full lines) and experimental (white circles) eye diagrams for (a) *r*S = 2.2 dB, (b) *r*S = 3.8 dB, and (c) *r*S = 6.9 dB. In all the cases *r*G = 4 dB.

Fig. 13. Simulated (full lines) and experimental (white circles) eye diagrams for (a) *rS* = 2.2 dB, (b) *rS* = 5.1 dB, and (c) *rS* = 6.9 dB. In all the cases *rG* = 5 dB.

A very good agreement between the simulated and experimental eye diagrams is observed in Figs. 12 and 13 for the two values of *rG* used, and similar results were observed for other values of *rG*.

The eye diagrams for the quaternary signal that propagated through 75 km of standard fiber are shown in Fig. 14. The agreement between experimental eye diagrams and numerical results is also very good. The experimental BERs are lower than 10-13 and 10-11 before and after transmission, respectively. In principle, it is possible to improve such values by using a narrower optical band-pass filter or by increasing the quaternary signal average power. Table 1 presents estimations for the BERs before and after propagation by using Eqs. (29).

Table 1. BER estimation before and after fiber propagation.

All-Optical Amplitude Multiplexing Through

Fiber Parametric Interaction Between Binary Signals 23

Fig. 15. Scheme illustrating the principle of operation of OAM-3F.

*ij P* :

straightforward task. If one detects power levels corrspeonding to 01

signal is, in principle, not possible because 00 10

following four power levels, *out*

by inspecting Fig. 15a or Eqs. (35).

The OBPF is centered at *ω+*; therefore, when the signal at *ω1* conveys a bit *i* (= "0" or "1") and the signal at *f2* carries a bit *j* (= "0" or "1"), the signal at the fiber output presents the

00 0 0 2

10 1 0 2

01 0 1 2

11 1 1 2

Since 00 10 0 *out out P P* , the output signal now is a 3-ASK one. Recovering information transmitted by the *ω2* signal from the generated ternary amplitude signal is a

*f2* clearly transmits a bit "1"; otherwise it transmits a bit "0". This may be promptly verified

On the other hand, recovering the pump information from the generated ternary amplitude

1 2 0 00 *out P kP P* (35a)

1 2 0 00 *out P kP P* (35b)

1 2 0 0 *out P kP P* (35c)

1 2 0 0 *out P kP P* (35d)

*out P* or 11

*out out P P* . This power level ambiguity implies

*out P* , the signal at

Fig. 14. Simulated (full lines) and experimental (white circles) eye diagrams: (a) before propagation, (b) after a 75 km standard fiber propagation. *rP* = 0.86 dB (*rG* 2.5 dB), *rS* = 4.4 dB.

The recovery of the original binary information from the quaternary-amplitude signal can also be performed optically by using some known techniques (Oda & Maruta, 2006), or by developing especial optical devices with the *S-* and *U*-shaped transfer functions (Fagotto & Abbade, 2010).

## **4. All-optical generation of ternary amplitude-shift keying signals**

In the previous section we showed how to multiplex two 2-ASK signals into a single 4-ASK one. To perform such operation, it was necessary to provide a power offset to both input binary signals. As a result, the lowest level power of the of the 4-ASK signal was not null, which degrades the signal BER performance.

In this section, we discuss two other techniques that multiplex two binary signals and generate a 3-ASK signal rather than a 4-ASK one. The first one, OAM-3F, is based on FWM; it requires a power offset on just one of the input binary signals. The second, OAM-3P, utilizes PA and holds for two OOK input binary signals. For the same average power, ternary-amplitude signals present lower BERs than quaternary ones. Besides, the 3-ASK signal lowest power level is null for both techniques; this also contributes to reduce the BER of the multiplexed signal and it is an important advantage for the techniques presented here. However, recovering information relative to two binary signals from a single ternary one requires the use of some special signal characteristics. All of these aspects are detailed below.

#### **4.1 Optical amplitude multiplexing through fiber four-wave mixing**

#### **4.1.1 Theory**

The notation employed here is the same utilized in Section 3.1. The principle of OAM-3F is illustrated in Fig. 15a. Two co-polarized OOK signals at *ω1* and *ω2* are coupled and propagated through an optical fiber with nonlinear and dispersion parameters appropriate for favoring FWM; however, it is assumed that the signal at *ω2* is an OOK, so 0 2*P* 0 0 .

Fig. 14. Simulated (full lines) and experimental (white circles) eye diagrams: (a) before propagation, (b) after a 75 km standard fiber propagation. *rP* = 0.86 dB (*rG* 2.5 dB), *rS* = 4.4

**4. All-optical generation of ternary amplitude-shift keying signals**

**4.1 Optical amplitude multiplexing through fiber four-wave mixing**

The recovery of the original binary information from the quaternary-amplitude signal can also be performed optically by using some known techniques (Oda & Maruta, 2006), or by developing especial optical devices with the *S-* and *U*-shaped transfer functions (Fagotto &

In the previous section we showed how to multiplex two 2-ASK signals into a single 4-ASK one. To perform such operation, it was necessary to provide a power offset to both input binary signals. As a result, the lowest level power of the of the 4-ASK signal was not null,

In this section, we discuss two other techniques that multiplex two binary signals and generate a 3-ASK signal rather than a 4-ASK one. The first one, OAM-3F, is based on FWM; it requires a power offset on just one of the input binary signals. The second, OAM-3P, utilizes PA and holds for two OOK input binary signals. For the same average power, ternary-amplitude signals present lower BERs than quaternary ones. Besides, the 3-ASK signal lowest power level is null for both techniques; this also contributes to reduce the BER of the multiplexed signal and it is an important advantage for the techniques presented here. However, recovering information relative to two binary signals from a single ternary one requires the use of some special signal characteristics. All of these aspects are detailed

The notation employed here is the same utilized in Section 3.1. The principle of OAM-3F is illustrated in Fig. 15a. Two co-polarized OOK signals at *ω1* and *ω2* are coupled and propagated through an optical fiber with nonlinear and dispersion parameters appropriate

2*P* 0 0 .

for favoring FWM; however, it is assumed that the signal at *ω2* is an OOK, so 0

dB.

Abbade, 2010).

below.

**4.1.1 Theory**

which degrades the signal BER performance.

Fig. 15. Scheme illustrating the principle of operation of OAM-3F.

The OBPF is centered at *ω+*; therefore, when the signal at *ω1* conveys a bit *i* (= "0" or "1") and the signal at *f2* carries a bit *j* (= "0" or "1"), the signal at the fiber output presents the following four power levels, *out ij P* :

$$P\_{\text{out}\_{+}}^{00} = k\_{+} P\_{1}^{0}(0) \Big[ P\_{2}^{0}(0) \Big]^{2} = 0\tag{35a}$$

$$P\_{out+}^{10} = k\_+ P\_1^1(0) \left[ P\_2^0(0) \right]^2 = 0 \tag{35b}$$

$$P\_{\text{out}+}^{01} = k\_+ P\_1^0(0) \left[ P\_2^1(0) \right]^2 \tag{35c}$$

$$P\_{out+}^{11} = k\_+ P\_1^1(0) \left[ P\_2^1(0) \right]^2 \tag{35d}$$

Since 00 10 0 *out out P P* , the output signal now is a 3-ASK one. Recovering information transmitted by the *ω2* signal from the generated ternary amplitude signal is a straightforward task. If one detects power levels corrspeonding to 01 *out P* or 11 *out P* , the signal at *f2* clearly transmits a bit "1"; otherwise it transmits a bit "0". This may be promptly verified by inspecting Fig. 15a or Eqs. (35).

On the other hand, recovering the pump information from the generated ternary amplitude signal is, in principle, not possible because 00 10 *out out P P* . This power level ambiguity implies

All-Optical Amplitude Multiplexing Through

1 2

power level is detected. In this case, one can write:

1

*BER P l P l P i P i P h P h*

01 10 11 01 01 10 11 01

*ii ii Q Q*

**4.1.2 Results and discussion**

parameters are:

signals are repeated.

Fiber Parametric Interaction Between Binary Signals 25

To estimate *BER1*, we assume that an EDC, such as the one illustrated in Fig. 2, is programmed to decide which bit was sent by the signal at *ω1* based only on the first non-null sample of the acquired *N* samples (assumption (c) above). In this case, the EDC interprets that the signal *ω1* sent a '0'-bit or '1'-bit when, respectively, the intermediate or the highest

> 01 10 01 10 11 01 11 01 01 10 01 10 11 01 11 01

where '*l*', '*i*', and '*h*' stand, respectively, for the lowest, intermediate, and highest amplitude levels, *P x* is the probability of occurrence of level *x* (*x*= '*l*', '*i*', and '*h*'), and *P x*

conditional probabylity that an error occurs given that level *x* (*x*= '*l*', '*i*', and '*h*') is detected. Although the theory presented above was derived for a signal with finite ER at *ω1* and an OOK signal at *ω2*, the proposed technique could also be applied for an OOK signal at *ω1* and a signal with finite ER at *ω2*. Also, the FWM component selected by the OBPF could be at *ω-*. Eqs. (35)-(38) should be adapted in all of the possible combinations. Nevertheless, the

Considering the system shown in Fig. 15b, Fig. 16 presents simulation results for bit sequences for the (a) signal at *ω1*= 193.1 THz with *r1*= 2 and *Rb1*= 2.5 Gb/s, (b) signal at *ω2*= 193.2 THz with *Rb2*= 10 Gb/s, and (c) signal at the OBPF output at *ω+*= 193.3 THz. Fiber

*0* = 1552.52 nm, *S0* = 0.017 ps/nm2/km, = 5.3 W-1km-1,

*L*= 3.0 km. The signal observed in Fig. 16c clearly represents a ternary-amplitude one. Moreover, the same power levels are obtained when the bits of the pump and the probe

general aspects treated here, like the power level optimization would still hold.

Fig. 16. Input signals at (a) *ω1* and (b) *ω2* and output signal at (c) *ω+*.

(38b)

= 0.2 dB/km, and

1 1 1 1 2 2 4 4

*ii ii ii ii Q QQ Q*

that when the lowest (null) power level is detected, the signal at *f1* may either convey a "0" or a "1"-bit. However, it is possible to solve this problem if the following assumptions are made:

$$R\_{b2} = NR\_{b1\text{ \textquotedblleft}}\tag{36}$$

where *N* is the same integer number considered in assumption (a), and

	- i. if only power levels 11 *out P* and 00 *out P* are detected, then the signal at *ω1* sent a "1-bit";
	- ii. if only power levels 01 *out P* and 00 *out P* are detected, then the signal at *ω1* sent a "0-bit";
	- iii. if any other combination of power levels is detected, then there is an error and information should be discarded.

The detection rules just described may be verified by simple inspection of Fig. 15b, where rules (a), (b), and (c) were considered for N= 4.

As stated before, power level optimization for optical multi-amplitude signals depends on the kind of the dominant noise and it is analyzed in (Walklin & Conradi, 1999). In case ASE noise is dominant, such optimal distribution must follow a quadratic law; for a ternaryamplitude signal, this means that power levels must be distributed following proportions of 0: 1: 4. Eqs. (35d) show that such distribution may be easily obtained by setting:

$$\frac{P\_{\text{out}\gets}^{11}}{P\_{\text{out}\gets}^{01}} = \frac{P\_1^1(0)}{P\_1^0(0)} = r\_1 = 4 \tag{37}$$

Therefore, power level optmization may be easily achieved when using this technique. In fact, OAM-3F is able to generate 3-ASK signals with arbitrary power level distributions, by solely setting the ER of the signal at *ω1*.

After inspecting Fig. 15b and following a reasoning similar to the one considered in Section 3.1, it is easy to conclude that:

$$BER\_2 = Q \left( \frac{i\_{01} - i\_{00}}{\sigma\_{01} + \sigma\_{00}} \right) \tag{38a}$$

that when the lowest (null) power level is detected, the signal at *f1* may either convey a "0" or a "1"-bit. However, it is possible to solve this problem if the following assumptions are made: a. The line code used by the signal at *ω2* guarantees a maximum of (*N*-1) bits 0 in a row, *i.e.* in a sequence of *N* bits there is at least one bit "1". In fact, this characteristic is implemented by most practical line codes to minimize the chance of the receiver loosing synchronism to the transmitter. For example, the line code utilized in Gigabit Ethernet standard, called 8B/10B, assures that no bit sequence presents more than four bits "0"

*R NR b b* 2 1 , (36)

*out P* are detected, then the signal at *ω1* sent a "1-bit";

*out P* are detected, then the signal at *ω1* sent a "0-bit";

(37)

(38a)

b. the signal bit rate at *ω2*, *Rb2*, is higher than the signal bit rate at *ω1*, *Rb1*:

the signal at *ω1* are the following. In a sequence of *N* samples:

*out P* and 00

*out P* and 00

where *N* is the same integer number considered in assumption (a), and

c. to recover information conveyed by the signal at *ω1*, such a signal is oversampled at a rate *Rb2*. Because of (b), this oversampling process establishes that *N* samples of the signal at *ω1* are obtained for each of its bits. However, following (a), at least in one of these samples, the signal at *ω2* shall transmit a "1"-bit. Therefore, the detection rules for

iii. if any other combination of power levels is detected, then there is an error and

The detection rules just described may be verified by simple inspection of Fig. 15b, where

As stated before, power level optimization for optical multi-amplitude signals depends on the kind of the dominant noise and it is analyzed in (Walklin & Conradi, 1999). In case ASE noise is dominant, such optimal distribution must follow a quadratic law; for a ternaryamplitude signal, this means that power levels must be distributed following proportions of

>

0

Therefore, power level optmization may be easily achieved when using this technique. In fact, OAM-3F is able to generate 3-ASK signals with arbitrary power level distributions, by

After inspecting Fig. 15b and following a reasoning similar to the one considered in Section

4

*r*

01 00

01 00

0

0: 1: 4. Eqs. (35d) show that such distribution may be easily obtained by setting:

*out out*

11 1 1 1 01 0 1

*P P*

*P P*

2

*i i BER Q*

in a row of ten bits;

i. if only power levels 11

ii. if only power levels 01

information should be discarded.

rules (a), (b), and (c) were considered for N= 4.

solely setting the ER of the signal at *ω1*.

3.1, it is easy to conclude that: