CALIFORNIA STATE UNIVERSITY, NORTHRIDGE Dispersion Managed Soliton a Graduate Project Submitted in Partial Fulfillment of the Re

CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

Dispersion Managed Soliton

A graduate project submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering

Murali Karthik Kothapalli

December 2017

The Graduate Project of Murali Karthik Kothapalli is approved:

Professor Xiaojun Geng Date

Professor Jack Ou Date

Professor Nagwa Bekir, Chair Date

California State University, Northridge

Acknowledgements

The completion of this project would not have been possible without the continuous support and feedback from Dr. Nagwa Bekir, Dept. of Electrical and Computer Engg., and it was my pleasure working under her guidance. I would also like to express my gratitude to the committee members Dr. Geng Xiaojun and Dr. Jack Ou for their academic and moral support throughout the project.

Last but not the least I would like to thank everyone who has directly or indirectly contributed in making the project a success.

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Table of Contents

Signature Page………………………………………………………………...... ii Acknowledgements……………………………………………………………………...…...... iii List of Figures………………………………………………………………………...…………...v Abstract………………………………………………………………….…………………….....vii 1. Introduction to Optical Soliton……………………………………………………………..….1 1.1. Introduction to Optical Soliton………………………...... 2 2. Evolution of an Optical Soliton: Mathematical Formulation…...... 3 2.1. Evolution of Optical Soliton: Mathematical Formulation………………………………...3 2.2. Nonlinear Schrodinger Equation……………………………...... …………....3 2.3. Bright Solitons…………………………………………………………………………...4 3. Losses inside a Fiber……………………………………………………………..…………….7 3.1. Losses inside a Fiber……………………………………...……………………………....7 3.2. Group Velocity Dispersion……………………………...………………………………..7 3.3. Self-Phase Modulation……………………………………...……………….……………9 4. Communication Based on Soliton…………………………………………….……………...12 4.1. Communication Based on Soliton……………………………………………………….12 4.2. Transmission of Information Using Solitons…………………………….………………12 4.2.1. Initial Frequency Chirp…………………………………………..……………...12 4.3. Soliton Transmitters……………………………………...……………….……………..13 4.4. Interaction of Solitons…………………………………………………………………...15 5. Simulation and Results………………………………………………………..……………...18 5.1. System Design…………………………………………………………...……………...18 5.2. Results………………………………………………………………….…...…………...19 5.2.1. Group Velocity Dispersion Analysis………………………...……...……………19 5.2.2. Self-Phase Modulation Analysis…………………………...………...…………..27 5.2.3. BER (Bit Error Rate) Analysis……………………………...…………………… 30 5.2.4. GVD and SPM combined effect……………………………...……...…………...31 6. Conclusion

List of Figures

Figure 2.1: First order and Third order Soliton pulse for one soliton period………………………………………………..………………………………………….…5 Figure 2.2: Pulse evolution of a fundamental soliton for a distance range of 0 to 10 km………………………….…………………………………………………………..………….6 Figure 3.1: Frequency chirp introduced by SPM…………………...... ……………...10 Figure 4.1: Figure shows the fraction of bit slot occupied by a Soliton……………………………………………………..………………………....…………..12 Figure 4.2: Evolution of a chirped input pulse into a fundamental soliton for N = 1 and C = 0.5………………………………………………………………………………………………...13 Figure 4.3: Figure shows a) Schematic b) Soliton pulse source…………………………………14

Figure 4.4: Soliton interaction for test values of relative amplitude, r and relative phase, 휃……16 Figure 5.1: Circuit Diagram to demonstrate soliton pulse propagation…………………....….....18 Figure 5.2: Layout parameters for the soliton system………………………………..…………..19 Figure 5.3: Optical fiber nonlinear properties in an optical fiber……...... 20 Figure 5.4: Dispersion parameters in an optical fiber………………………………….………...20 Figure 5.5: Pulse broadening due to GVD for fiber length of 5 km…...... 21 Figure 5.6: Pulse broadening due to GVD for fiber length of 10 km………………...………….21 Figure 5.7: Pulse broadening due to GVD for fiber length of 15 km…………………...……….22 Figure 5.8: Pulse broadening due to GVD for fiber length of 20 km…………………...……….22 Figure 5.9: Pulse broadening due to GVD for fiber length of 25 km…….……………...... …….23 Figure 5.10: Pulse broadening due to GVD for fiber length of 30 km...... 23 Figure 5.11: Pulse broadening due to GVD for fiber length of 35km…………………...………24 Figure 5.12: Pulse broadening due to GVD for fiber length of 40 km….……………...……...... 24 Figure 5.13: Pulse broadening due to GVD for fiber length of 50 km…………………..……....25 Figure 5.14: Full Width Half Maximum (FWHM) vs Fiber Length (Practically evaluated)……………………………………………………...... 26

Figure 5.15: Full Width Half Maximum (FWHM) vs Fiber Length (Theoretically evaluated)……………………………………………………………………...…………………26 Figure 5.16: Effects of SPM for a fiber length of 15 km………………………..……………….28 Figure 5.17: Effects of SPM for a fiber length of 30 km………………………..……………….28 Figure 5.18: Effects of SPM for fiber length of 45 km…………………………..……………....29 Figure 5.19: Effects of SPM for fiber length of 60 km…………………………………...... 29 Figure 5.20: min BER vs fiber length when only GVD is acting inside the fiber……………………………………………………………………………...... 30 Figure 5.21: min BER vs fiber length when only SPM is acting inside the fiber……………………………………………………………………………...... 31 Figure 5.22: Soliton in time domain after travelling 5 km……………………...... 32 Figure 5.23: Soliton in time domain after travelling 15 km………………………...... 32 Figure 5.24: Soliton in time domain after travelling 30 km…………………………………...... 33 Figure 5.25: Fiber Length vs Full Width Half Maximum when both SPM and GVD are acting…………………………………………………………………………………………...... 33

ABSTRACT

Dispersion Managed Soliton

Murali Karthik Kothapalli

Master of Science in Electrical Engineering

This project demonstrates the design and performance analysis of an optical soliton in long distance communication systems. The soliton systems are stable and lossless, which are preserved throughout the length of the communication channel. These solitons do not spread along the link length and is reliable as compared to non-soliton communication systems. The soliton power levels are always maintained to a certain limit. The solitons are formed because of an interplay between the Group Velocity Dispersion and Self-Phase Modulation, which individually have depreciating effects on the propagating pulse. This report also demonstrated on how a sech pulse is converted to a soliton for propagation. When two solitons interact with each other during the propagation, they still would maintain their shape and power resulting in an elastic collision.

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1. Introduction

1.1 Introduction to Optical Soliton: Optical fiber communication is a method by which we can transmit information using light pulses through an optical fiber. This method can be widely used in long-haul communications like telephone, Internet, Cable TV etc. The light travels as an electro- magnetic wave inside the optical fiber, which is modulated and serves as a carrier for carrying information. Optical fibers ae used instead of the electric cables like the copper wires, co-axial cables etc. This has changed the way of communication in telecommunication industry by providing low attenuation and interference. The light pulses travelling inside the fiber gradually broaden and peak power decreases with time and distance, and eventually overlap with other pulses and makes it impossible for the receiver input to distinguish between the pulses. This can be called as the Group Velocity Dispersion (GVD). Thus, dispersion can affect the transmission capacity and bandwidth. This dispersion can be a major problem for systems which need high bit rate and for long- distance communication systems. Optical Soliton which can cancel the dispersive effects offers a better solution for this problem.

Optical solitons are solitary wave packets which can reinforce themselves to effectively balance the dispersive effects, also known as the GVD, with the nonlinear dispersive effects as it travels through the medium inside the optical fiber. Thus, optical soliton can retain its shape as it travels with time and constant velocity. The nonlinear dispersive effect also called as the Self Phase Modulation (resulting from Kerr nonlinearity) changes the frequency of the wave by causing a time-varying phase shift which directly is dependent on the intensity of the light pulse. Also, the refractive index is dependent on the intensity of the light pulse. The leading edge has a higher refractive index whereas the tail edge has the lower refractive index which eventually results in phase-shift. As we know the derivative of phase shift is frequency, any change in phase shift results in a frequency change. Thus, a soliton does not change its shape because this nonlinear property suppresses the broadening of the pulse eliminating the Group Velocity Dispersion and Inter Symbol Interference (ISI). For detailed description refer to Chapter-3.

The concept of dispersion managed soliton has been a major area of research because it can be practically available in many optical communication devices and systems. In terms of nonlinear optical systems, there are two types of solitons: temporal solitons and spatial solitons both of which are the result of perfect balance between the negative Group Velocity Dispersion (GVD) and positive Self Phase Modulation (SPM). Due to its reliability as the analysis looks the same for any distance along the optical fiber, due to this self-reinforcing parameter the optical solitons can be a resourceful solution for the limitations on date-rate and capacity. Hence optical solitons are a breakthrough in optical

1 communication systems and proves to be a possible solution for long distance communications and high data rate required optical systems.

To fully understand the optical soliton solitary waveform, we need to study the soliton formation and flow inside the optical fiber when dispersion and nonlinear parameters are acting within the fiber. The mathematical understanding of a soliton using the Nonlinear Schrodinger equation (NLS equation) is explained in Chapter-2. The soliton transmitters, interaction between adjacent solitons and the transmission of information using solitons is explained in Chapter-4. The Chapter-5 contains the Group Velocity Dispersion analysis, Self-Phase Modulation analysis, BER analysis which explains the effects of these parameters on a pulse travelling inside a fiber.

2. Evolution of Optical Soliton

2.1 Evolution of an Optical Soliton: Mathematical Formulation: The pulse of a soliton can withstand a collision like particles in addition to travelling undisturbed through the fiber. The advent of soliton started in an experiment conducted during 1988 where the loses of the fiber are compensated by the Raman amplification technique. Thus, rigorous experiments which leads to a progress in soliton practical usage in modern communication optical systems. This material is taken from reference [3] page 404-409, section 5.3 and section 2.3.

2.2 Nonlinear Schrödinger Equation: To design an Optical fiber, we need to consider all the possible non-linear effects, the phenomenon which can be a major limiting factor are the non-linear effects especially the SPM (self-phase modulation). The interplay between the non-linear effects and the GVD (dispersive effects) is very important for the propagation of the soliton. The equation which can govern the effects of SPM and GVD is well established in the NLS equation short form for nonlinear Schrödinger equation. The equation can be given as, 2 3 ∂A + ⅈβ2 ∂ A − β3 ∂ A = − α A + i̇γ|A|2A (2.2.1) ∂z 2 ∂t2 6 ∂t3 2 The pulse envelope is represented by A (z, t). The alpha term (α) is included to represent the fiber losses. The nonlinear parameter is given by γ which is written in terms of nonlinear refractive index 푛2 as,

γ = 2πn2 (2.2.2) λAeff

The term β2 is the second order dispersion effect whereas β3 is the third order dispersion effect. The wavelength is given by λ and effective core area of the optical fiber by Aeff. To normalize the equation (2.1.1), it is firstly very important to set the values α and β3 to be zero (these parameters will not be included or discussed in this paper). The following assumptions are to be made,

τ = t , ξ = z , U = A (2.2.3) T0 LD √P0

To is called the pulse width, Po is called the pulse peak power and LD is called the length of dispersion. Now equation (2.1.1) can be written as,

2 ⅈ ∂U − s ∂ U + N2|U|2U = 0 (2.2.4) ∂ξ 2 ∂τ2

Depending on whether β2 is + GVD (called normal GVD) or a - GVD (called anomalous GVD) discussed in Chapter-3, the equation s = sgn (β2) can be +1 or -1. The conceptual

3 understanding of the parameter N is discussed later, while the equation representing N can be deduced to be,

2 2 N = γLDPO = PO∂T0 ∕ |β2| (2.2.5) The best technique to solve the NLS equation is to use the inverse scattering method because it is a special case of nonlinear partial differential equations precisely talking in soliton terms. Many published materials are available for solving the NLS equation using the inverse scattering method [4]. This method will not be discussed in this report. The best form of soliton (soliton in pulse form) can only obtained using the anomalous GVD whereas for the normal GVD there is always a constant dip in the solutions for the soliton dynamics called as the dark solitons. In this chapter much consideration is given to the pulse-like form of a soliton called the bright solitons.

2.3 Bright Solitons (Pulse-like Solitons): By considering the case of an anomalous GVD where the equation becomes s = -1. The NLS equation can be remodified to a canonical form by introducing the renormalized amplitude, u = NU. The equation can thus be written as,

2 ⅈ ∂u + 1 ∂ u + |u|2u = 0 (2.3.1) ∂ξ 2 ∂τ2 Like it is said before, nonlinear partial differential equations can be solved by using the inverse scattering method. So, the equation thus transforms in the form,

u(0, τ) = N sec h (τ) (2.3.2) The equation above represents a pulse which is launched into an optical fiber and remains undistorted or retains its shape throughout the fiber when N = 1 thus can also be called the fundamental soliton. For values of N greater than 1, the pulse follows a periodic pattern called the higher-order solitons. The input shape for the higher-order solitons are recovered gradually at some point along the fiber, the period for the soliton given by zo is defined as the distance that a higher-order soliton covers to recover the original input pulse and it is shown by,

2 π π T0 z0 = LD = (2.3.3) 2 2 |β2| Only the fundamental soliton is of much importance because it does not change its frequency and remain un-chirped throughout the propagation along the optical fiber. The plots corresponding to the solitons when N = 1 and N = 3 for one soliton pulse period are shown below,

Figure 2.1: These graphs represent the first-order (top left and bottom left) and third-order (top right and bottom right) Soliton Power or intensity of pulse |u(ξ, τ)|2 and frequency profile (short and sharp pulses) for one Soliton period. These pictures were taken from Reference [3] Page 407. The equation (2.2.1) can be directly solved to obtain the solution for a bright fundamental soliton without using the inverse scattering method. The fundamental procedure is that by assuming the soliton pulse envelope amplitude to be,

u (ξ, T) = V (τ) exp [ⅈϕ(ξ)] (2.3.4)

It is very important to keep V independent of ξ and also the phase ϕ dependent on ξ to obtain a fundamental pulse that does not change its shape as it propagates along the fiber length. By substituting the equation (2.2.4) in equation (2.2.1) and solving for ϕ and V, their respective equations are obtained which are given by,

ϕ (ξ) = 퐾ξ and V(τ) is represented by a nonlinear differential equation,

2 ⅆ V = 2V (K − V2) (2.3.5) ⅆτ2

By multiplying 2 (ⅆV) and integrating with respect to τ, the above equation would transform ⅆτ into,

2 (ⅆV) = 2kV2 − V4 + C (2.3.6) ⅆT

The value of C is determined to be 0 by considering the first boundary condition, where V and ⅆv ∕ ⅆτ should diminish as |τ| approaches to infinity. The other boundary condition would determine the value of K = ½ by considering V = 1 and ⅆv ∕ ⅆτ = 0. Now the value of ϕ comes out be ξ/2. Thus, by integrating equation (2.2.6), the equation obtained is V (τ) = sech (τ). The resulting pulse envelope thus takes the form,

u (ξ, τ ) = sec h (τ) exp (ⅈξ ∕ 2) (2.3.7) This pulse envelope thus can be substituted in the NLS equation directly and apply the integration without the use of inverse scattering method. Thus, to summarize the above discussion, the pulse has a phase deviation of ξ ∕ 2 and the pulse envelope does not change its amplitude and remains constant throughout the propagation inside the optical fiber by exactly compensating the nonlinearities with the effects of dispersion (GVD). Thus, it can be concluded from the above mathematical solutions that the pulse shape should be a “sech”. Hence, these properties derived above can make the soliton propagation a possible worthy candidate in the field of optical communication systems. The fundamental soliton (N=1) has to be created such that it has to maintain constant shape and peak power throughout the fiber length. It is possible for a soliton to regain its shape and peak power even though it initially deviates from the ideal situation. For ξ >> 1, the input pulse tends to adjust its shape and width as well to retain the original fundamental soliton. The figure shown below provides a simulated graph for an input pulse gradually evolving into a fundamental soliton provided N = 1,

Figure 2.2: The above picture shows the input pulse evolving into a fundamental soliton by maintaining its shape, power and width for N = 1 and ξ ranges from 0 to 10. This picture is taken from reference [3] page 408.

3. Losses Inside a Fiber

3.1 Losses inside a fiber: The discussion in the previous chapter was to get an insight into the mathematical formulation of a soliton propagation. In this chapter, there will be a better understanding of the type of losses which can lead to the evolution of a soliton. There are many losses inside a fiber which can distort the light propagation inside an optical fiber. But, two of the losses are a necessity on the context of this report. The interchange between Group Velocity Dispersion (GVD) and Self Phase Modulation (SPM) causes a soliton which makes the discussion of these two types of losses very important. The first part of the chapter is about GVD and will continue to discuss about SPM in the later sections of this chapter.

3.2 Group Velocity Dispersion: As the light pulse travels along the fiber length L, the pulse is composed of many spectral components each with a particular frequency 휔. Hence each spectral component of certain frequency arrives at the end of a fiber with a definite time delay given by 푇 = 퐿 , 휈푔 where 휈푔 is called as the group velocity. The group velocity is given by,

휈 = 휕휔 (3.2.1) 푔 휕훽 where 휔 is the frequency related to a specific spectral component and 훽 is called as the propagation constant. More details about the group velocity formula is given in section 2.3.3 of reference [2].

Considering 훽 = 푛̅푘 = 푛̅ . 휔 and substituting in equation (3.1.1), the group velocity can be 0 푐 written as,

푐 휈푔 = (3.2.2) 푛̅푔 where 푛̅푔 is called as the group index and be represented as,

푛̅푔 = 푛̅ + 휔(ⅆ푛̅ ∕ ⅆ휔) (3.2.3) As different spectral components arrive at the output at different timings instead of arriving simultaneously due to the dispersion along the propagation distance which can lead to the pulse broadening. Thus, one can infer that frequency is dependent on the group velocity leading to the above-mentioned pulse broadening. The pulse broadening resulting inside an optical fiber of length L considering the spectral width 훥휔 is governed by,

ⅆ푇 ⅆ 퐿 ⅆ2훽 훥푇 = 훥휔 = ( ) 훥휔 = 퐿 2 훥휔 = 퐿훽2훥휔 (3.2.4) ⅆ휔 ⅆ휔 휈푔 ⅆ휔

2 The Group Velocity Dispersion(GVD) parameter is given by the term ⅆ 훽, which can ⅆ휔2 define the extent by which the broadening of the optical pulse takes place inside the optical fiber.

Sometimes the spectral width 훥휔 from a source is written in terms of the wavelength as 훥휆 in optical communication. Thus, by substituting 훥휔 = (-2휋푐) 훥휆 in the above equation 휆2 (3.1.4) turns out to be,

훥푇 = ⅆ ( 퐿 ) 훥휆 = DL 훥휆 (3.2.5) ⅆ휆 휈푔 Where D is called as the dispersion parameter and can be given by,

ⅆ 1 2휋푐 D = ( ) = (- 2 ) 훽2 (3.2.6) ⅆ휆 휈푔 휆 The pulse broadening can affect the bit rate(B) which can be roughly calculated by utilizing the condition B 훥푇 < 1 and the value of 훥푇 is taken from equation (3.1.5). The condition for the dispersion effects on the bit rate B is given by,

BL|D| 훥휆 < 1 (3.2.7) For the silica single mode fibers using a semiconductor laser source, D is relatively small quantity near the wavelength region 1.3µm. The BL product for this optical communication system comes out be 100 (Gb/s)-km (for 2-4 nm wavelength range 훥휆) and can exceed this value. When the wavelength range 훥휆 is reduce to 1nm from the above mentioned 2-4 nm the BL product comes out be 1(Tb/s)-km. This clearly shows that dispersion effects have a considerable influence on the bit rate B.

The frequency is dependent on the mode index 푛̅ which is the influence behind frequency dependence of dispersion parameter D. Thus D, by using the equation (3.1.3) can be written as,

2 퐷 = − 2휋퐶 ⅆ ( 1 ) = −2휋 (2 ⅆ푛̅ + 휔 ⅆ 푛̅) (3.2.8) 휆2 ⅆ휔 휈푔 휆2 ⅆ휔 ⅆ휔2 Dispersion D can be classified into two types, material dispersion and waveguide dispersion. The formulation on how to deduce the parameter D into two components DW and DM is shown in section (2.2.3) of reference [3].

The material dispersion DM and waveguide dispersion DW are given by,

The thorough discussion of these two dispersion parameters is shown in section (2.3.2) and section (2.3.3) of reference [3].

3.3 Self-Phase Modulation: Considering single-channel light wave systems, the nonlinear effect which can greatly affect the system performance is Self-Phase Modulation. All the materials exhibit nonlinear effects at elevated input light intensities and refractive index increases with the intensity. Thus, refractive index is dependent on power of the source. To expose the nonlinear effects, modifying the refractive indices of the fiber’s core and cladding would be a necessity. The modified refractive indices are given as,

1 푛푗 = 푛푗 + 푛̅2(푃 ∕ 퐴푒푓푓) j = 1,2 (3.3.1)

The term 푛̅2 is given by nonlinear index coefficient, where P is known as the Optical Power and 퐴푒푓푓 is given by effective mode Area. By doping the core by some fraction the value -20 2 of 푛̅2 comes out to be 2.4 * 10 m /W. Thus, the nonlinear refractive index coefficient is -12 a relatively small value (< 10 ) because of a very small value for 푛̅2 at the Optical Power level of 1 milliwatt. Despite the very low values to these terms, they can considerably effect the optical fiber communication systems for very long distance optical fibers and can also lead to the concept of cross-phase modulation and self-phase modulation. The concept of cross-phase modulation will not be discussed in this report and is more concentrated on self-phase modulation. The propagation constant greatly changes and becomes dependent on power and can be given as,

′ 훽 = 훽 + 푘0푛̅2푃 ∕ 퐴푒푓푓 ≡ 훽 + 훾푃 (3.3.2) The most essential nonlinear parameter in the above equation is,

훾 = 2휋푛̅2 ∕ 퐴푒푓푓휆 (3.3.3)

The value of the above equation can range from 1 to 5 푊−1 ∕ 푘푚 majorly relying on the value of 퐴푒푓푓 and wavelength. The above term creates a nonlinear phase shift as the optical phase is linearly proportional with z. This nonlinear phase shift can be shown as,

퐿( ′ ) 퐿 ( ) 휙푁퐿 = ∫0 훽 − 훽 ⅆ푧 = ∫0 훾푃 푧 ⅆ푧 = 훾푃푖푛퐿푒푓푓 (3.3.4)

The term 푃(푧) determines the fiber losses and 퐿푒푓푓 (effective interaction length) is given by,

퐿푒푓푓 = [1 − 푒푥푝(−훼퐿)]⁄훼 (3.3.5)

Where 훼 accounts for the fiber losses and the term 푃푖푛 is deduced to be a constant. The term optical phase thus changes with time in the same fashion as that of the optical signal. As this nonlinear phase shift or modulation is self-induces, it is called as the Self Phase Modulation (SPM) thus leading to frequency chirping in the optical signals or pulses. This chirp in the frequency is dependent on the shape of the optical signal and is correlative to the derivative ⅆ푃푖푛 . ⅆ푡 The Figure (3.1) shows the SPM induced frequency chirp variations with time. The graphs are shown for the Gaussian (m=1) and super-Gaussian (m=3) pulses. These frequency chirps can affect the shape of the pulse and can also lead to additional pulse broadening with GVD. The SPM affects are accumulated at each amplifier and SPM induced phase can vary due to the multiple amplifiers present. For minimizing the impact of SPM on optical communication systems it is required that 휙푁퐿 ≪ 1.

Fig 3.1 Frequency chirp introduced by SPM.

Considering the maximum tolerable nonlinear phase shift as 휙푁퐿 = 0.1 and by replacing 1 퐿푒푓푓 by ⁄훼 for lengthy fibers thus we can produce the condition on the limit for the input peak power as,

푃푖푛 < 0.1훼 ∕ (훾푁퐴) (3.3.6) Using the optoelectronic regenerators, the SPM effects are a little concern if the input power satisfies the above condition i.e. when 푃푖푛 << 22 푚푊 and 푁퐴 = 1. But the SPM effects do accumulate over long distances (~1000 km) when amplifiers are placed periodically. To limit the SPM effects the above equation (3.2.6) gives a rough estimate of the peak power. According to the condition the peak power should not be above 2.2Mw for 10 amplifiers and the essential nonlinear parameter should be 훾 = 2 W-1 /km. For a light wave system operating for over 1000 km the average power should remain below 1 Mw for negligible SPM effects. To limit on the value for average power also depends on the

10 type of fiber the optical light propagates through the effective core area 퐴푒푓푓. The 퐴푒푓푓 should be close to 20 휇푚2 to restrict the SPM effects. This SPM effects when acting on an optical signal simultaneously with fiber dispersion effects leads to a much more important concept of a soliton. The nonlinear Schrodinger equation also shown as NLS equation includes the SPM effects acting on an optical signal inside an optical fiber. The equation for NLS equation is given by,

2 ∂A + ⅈβ2 ∂ A = − α A + i̇γ|A|2A (3.3.7) ∂z 2 ∂t2 2 For further details on NLS equation refer to section (2.1) of Chapter-2. The 훼 term in the above equation includes the losses in a fiber. Thus, we can conclude that NLS equation includes the required information to design a modern optical fiber communication system. The NLS equation must be solved numerically to resolve the nonlinear effects of SPM in long distance communication systems. The SPM effects can be considerably reduced to some extent by properly chirping the input signals. Also by using large effective area

(퐴푒푓푓) fiber, the nonlinear parameter γ can be reduced. The chirping of input signals has led to a new modulation format called as the chirped RZ (return to zero) or CRZ format.

4. Communication Based on Soliton

4.1 Communication based on Soliton: The usage of solitons in communication systems is different from that of a non- soliton system hence requires an entirely challenging communication system design. This entire chapter is dedicated to have a deep understanding of the soliton system from that of a conventional system.

4.2 Transmission of information using Solitons: The NLS equation governing the soliton, is only valid when the individual soliton pulses are well separated from each other. When the soliton propagation is used for information bits, the RZ digital line coding technique is the best solution and the soliton width occupies a small fraction of bit slot.

Figure 4.1: Above figure shows the fraction of bit space occupied by the soliton. 푇퐵 is defined as the width of each bit slot.

퐵 = 1 = 1 (4.1.1) 푇퐵 2푞0푇0 The above equation gives the relation between bit rate (B) and duration of each soliton 푧 (푇0). In Eq. (2.2.7), by making 휉 = = 0 in the solution for the fundamental soliton, we 퐿퐷 can obtain the input pulse characteristics required to kindle the fundamental soliton and thus one can obtain the power variation across the soliton pulse,

2 2 푃(푡) = |퐴(0, 푡)| = 푃0 푠푒푐 ℎ (푡 ∕ 푇0) (4.1.2)

From Eq. (2.1.5), the peak power (푃0) is obtained by making N = 1 as shown below,

2 푃0 = |훽2| ∕ (훾푇0 ) (4.1.3)

The 푇0 can be written in terms of the FWHM (Full Width Half Maximum) as,

푇푠 = 2푇표푙푛(1 + √2) = 1.76푇0 (4.1.4) 4.2.1 Initial Frequency Chirp: For the propagation of a fundamental soliton, the input source should produce a pulse that has a ‘sech’ description and must be chirp-free. But practically all the input pulses do have a frequency chirp which can harm the balance between SPM and GVD and

12 hence the soliton propagation. By studying the input amplitude, 푢(0, 휏) = 푠푒푐ℎ(푇) 푒푥푝(−𝑖퐶휏2 ∕ 2), the effects of frequency chirp in the input signal can be observed. Considering the case of N = 1 and C = 0.5, the soliton evolution can be shown as shown below,

Figure 4.2: Evolution of a chirped input pulse into a fundamental soliton for N = 1 and C = 0.5. An input pulse initially is compressed by large due to the positive chirp despite the nonexistence of nonlinear effects. As the distance grows the pulses gradually spread from the dominant peak and thus the main peak transforms into a soliton after travelling a distance 휉 > 15. Identical trends are observed for negative values of C. The soliton formation and stability is only possible for small values of C and when exceeds a critical value Ccrit the input pulse does not grow into a soliton. One can derive the critical value of C from the inverse scattering method and is usually dependent on the value of N and phase factor. Also, the chirp produced initially must be minimized because even though the initial chirp may not be detrimental for values of C less than critical value, the energy can be spread as dispersive waves during the journey of a soliton evolution. For the above case where N = 1 and C = 0.5, around 82% of the input pulse energy is transformed into a soliton and this percentage decreases to 62% for C = 0.8.

4.3 Soliton Transmitters: The soliton transmitters which can produce optical input signal which is free from frequency chirping with higher repetition rates and shape close to “sech”. Fiber Losses can be compensated using the erbium-doped fiber amplifiers (EDFAs) and the operating wavelength should be around 1.5 µm where the fiber losses are minimum. The first experiments on soliton transmitters used the gain switching technique for generating input pulses in the range of 20-30 picoseconds duration. These pulses are generated when the laser is biased below threshold and then periodically boosting it high above threshold. The pulses are chirped initially by using this technique. By referring to section () of chapter-3, one can make the pulses chirp-free by choosing an optical fiber

13 with positive GVD or normal GVD and the chirp parameter negative (C < 0), the pulses get compressed. Another technique that can be used to produce optical pulses is mode locking, thus the transmitters are called as mode-locked semiconductor lasers. The lasers are modulated at a specific frequency, which is the difference of two adjacent longitudinal modes. By varying the cavity length, one can control the modulation frequency. Most lasers have a compact cavity length (< 0.5 mm), typically producing a modulation frequency of 50 GHz. Designing an external cavity can help minimizing the modulation frequencies. The external cavity is designed by splicing a chirped fiber grating to the pigtail joined to the transmitter and provides wavelength stability to around 0.1 nm. This setup with help of a thermoelectric heater is used for tuning and mode locking of the input source to a specific modulation frequency from a wide range of frequencies. Semiconductor laser sources have drawbacks which root from the hybrid nature of it. Hence an alternative solution is the Mode-Locked fiber lasers. These lasers still should rely on a semiconductor laser. Active mode locking requires that the spacing between two longitudinal modes should be of higher order entirely due to long cavity lengths (> 1 m).

Such optical sources use a intracavity LiNbO3 modulator for transmission of solitons. An optical amplifier of semiconductor nature can be used for mode locking actively. This technique can produce pulses of 10 picoseconds duration and repetition up to 20 GHz.

Figure 4.3: Above figure shows the a) Schematic b) Soliton pulse source. Other technique involves introducing a Continuous Wave (CW) beam into the optical fiber with weak sinusoidal modulation. Thus, the combined SPM, GVD in a dispersion- decreasing fiber transforms the weakly modulated beam into very short soliton pulse train.

The repetition of these soliton pulses is represented by the initial frequency of sinusoidal modulation. This technique can be useful for producing ultrashort soliton pulses with higher repetition rates. This can be attributed to the dispersion profile of the fiber which can be created by attaching low and high dispersion fibers together or in other words by beating two optical pulses this atmosphere can be created. The other term for describing this profile is comb-like dispersion profile. By modulating the output of a DFB laser (Two distributed Feedback laser) with Mach-Zehnder modulator produces a 3-ps soliton train around 40 GHz. The DFB semiconductor laser, followed by an optical BPF (Band Pass Filter) modulated the phase of a pulse thus producing frequency modulated (FM) sidebands around the carrier frequency. Depending on filtering either one or two sidebands, one can use this as single wavelength or a dual-wavelength source. The Mach-Zehnder modulator operated using the NRZ pulse stream, converts a DFB laser CW output into optical RZ bit stream. Although these pulses do have a “sech” profile of a soliton, they are only considered because of the soliton formation proficiency of a fiber. 4.4 Interaction of Solitons: Following the discussion on NLS equation in the previous chapters, we can deduce that NLS equation is only satisfied by one soliton pulse. As the soliton only occupies only a small fragment of the pulse width TS, which is one of the major parameters considered for designing a soliton system. Hence, packing multiple solitons tightly can lead to a major disturbance among the neighboring solitons. Thus, a new phenomenon of soliton interactions can be studied in this section. The NLS equation, by considering the input amplitude as a combination of two soliton pulses as shown below,

푢(0, 휏) = 푠푒푐ℎ(휏 − 푞0) + 푟 푠푒푐ℎ[푟(휏 + 푞0)] 푒푥푝(푖̇휃) (4.4.1) We consider few of the terms in relative to both the solitons, here r is the relative amplitude,

휃 is the relative phase and 2푞0 is the relative separation between the two solitons. Considering the following cases,

Figure 4.4: Soliton interactions for different values of relative amplitude, r and relative phase, 휃.

All the above cases of soliton interactions intensely depend on r and 휃 which proves that these two parameters play a major role. For two solitons having equal amplitude and in- phase, they collide throughout the length of the fiber periodically. At 휃 = 휋 ∕ 4, initially they attract each other and eventually gets separated from each other. At 휃 = 휋 ∕ 2, the two solitons ward off from each other even more and the separation between them grows as they travel along the fiber. All the above cases are unacceptable because the relative phase changes as the two solitons propagate and can lead to jitter. To overcome this problem one solution might be to increase the relative spacing 푞0 between the two solitons. As the relative spacing is kept sufficiently large, the soliton interaction decreases and would not disrupt each other and both the solitons retain their initial position till the end of a fiber which is desirable. The inverse scattering method can be used to study the soliton interactions based on the initial value of 푞0. Considering the case of r = 1 and 휃 = 0, the separation between the neighboring solitons at any distance 휉 is shown below as,

2 푒푥푝[2(푞푠 − 푞0)] = 1 + 푐표푠[4휉 푒푥푝(−푞0)] (4.4.2) The spacing between the adjacent solitons vary as they propagate along the fiber with a period given by,

휉푝 = (휋 ∕ 2) 푒푥푝(푞0) (4.4.3)

If 휉푝퐿퐷 ≫ 퐿푇, the solitons that are neighboring to each other drift from each other by a very minute value hence the concept of soliton interaction becomes irrelevant. By using 2 −1 퐿퐷 = 푇0 ∕ |훽2| and 푇0 = (2퐵푞0) , the above condition can be re-written as,

2 휋 푒푥푝(푞0) 퐵 퐿푇 ≪ 2 (4.4.4) 8푞0 |훽2|

The drawback of using large values for 푞0 is that it can greatly affect the bit rate B of the soliton system. This can be illustrated by the above equation where the GVD parameter 훽2 2 = -1 ps /km. The bit rate B comes out be 10 Gb/s when the initial soliton spacing is 푞0 = 6. This drawback can be greatly mitigated by using different relative amplitudes for the neighboring solitons. The soliton spacing does not vary by a great deal (less than 10%) when the relative amplitudes contradict by 10% and initial spacing 푞0 = 3.5. Although the peak powers may deviate by a minute amount, but this variation may not harm the communication using a soliton system.

5. Simulation and Results [6]

5.1 System Design: Optiwave OptiSystem 10 can be used to study the effects of Group Velocity Dispersion (GVD), Self-Phase Modulation (SPM) and the formation of a fundamental soliton in an optical fiber. The circuit used for the analysis is shown in Figure 5.1. The circuit consists of three main elements: Transmitter, Circuit and Receiver. The transmitter circuit consists of User Defined Bit Sequence Generator and Optical Sech Pulse Generator. A sech pulse is considered an ideal pulse for the input to observe the soliton propagation. The channel consists of an Optical Fiber, the characteristics of which can be changed accordingly. The receiver unit consists of Optical Receiver along with three kinds of analyzers: Optical Spectrum Analyzer(OSA), Optical Time Domain Visualizer(OTDV) and BER Analyzer.

Figure 5.1: Circuit Diagram: to demonstrate the soliton propagation.

OSA and OTDV are used before and after the channel to observe the input and output pulses in the frequency and time domain. The layout parameters for the above-mentioned circuit to observe the soliton propagation are shown in Figure 5.2. The bitrate of the above optical system is 40 Gb/s and a 16- bit sequence length considering 128 samples for each bit. The BER analyzer is added to the circuit to get the number of errors occurred that may have distorted the input sequence due to various factors like noise, dispersion and inter symbol interference.

Figure 5.2: Layout parameters for the system.

5.2 Results: By individually considering the effects of GVD and SPM on the optical pulse for communication, one can observe the soliton evolution. As explained in the previous chapters, soliton propagation is an interplay between the Group Velocity Dispersion and Self Phase Modulation.

5.2.1 Group Velocity Dispersion Analysis: To study the effects of GVD on the optical pulse, the nonlinearities tab in the optical fiber component properties is un-checked as shown in Figure 5.3. The Group Velocity Dispersion in the Dispersion tab is checked and the Frequency domain parameter is checked as shown in Figure 5.4.

At 40 Gb/s, the bit slot is determined as 25 ps and hence the 푇퐹푊퐻푀 is 12.5 ps. The parameter 푇0 can be calculated by considering the relation between 푇0 and 푇퐹푊퐻푀 for sech pulses.

푇0 = 푇퐹푊퐻푀/1.763 and hence, 푇0 = 7.0902 ps

Figure 5.3: Nonlinearities in an optical fiber.

Figure 5.4: Dispersion parameters in an optical fiber.

The dispersion length 퐿퐷 is calculated as shown below, 2 퐿 = 푇0 = 2.5135 km (5.1) 퐷 |훽| By changing the optical fiber length, the dispersive effects acting on the optical pulse during the propagation for each fiber length can be studied. The pulse broadening for different fiber lengths beginning from 5 km to 50 km fiber lengths are shown below,

Figure 5.5: Pulse broadening for fiber length of 5 km.

Figure 5.6: Pulse broadening for fiber length of 10 km.

Figure 5.7: Pulse broadening for fiber length of 15 km.

Figure 5.8: Pulse broadening for fiber length of 20 km.

Figure 5.9: Pulse broadening for fiber length of 25 km.

Figure 5.10: Pulse broadening for fiber length of 30 km.

Figure 5.11: Pulse broadening for fiber length of 35 km.

Figure 5.12: Pulse broadening for fiber length of 40 km.

Figure 5.13: Pulse broadening for fiber length of 50 km.

As shown above, the width of the pulse changes with distance z, the extend of the pulse broadening due to group velocity dispersion can also be evaluated using the formula, 1 2 푇(푧) = [1 + (푧 ∕ 퐿퐷) ]2 푇0 (5.2)

After travelling a distance z = 퐿퐷, the pulse width broadens by a factor of √2.

90 85 25.13, 85 80 75 22.6, 76 70 20.1, 68 65 17.6, 65 60 15.1, 60 55 12.57, 54 50 10.1, 50 45 40 7.54, 41 35 30 5, 27 25 2.51, 22 20 15 0, 12 10 Full Width Half Maximum in Maximum (FWHM) ps Half Width Full 5 0 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 Fiber Length in km

Figure 5.14: Full Width Half Maximum(FWHM) vs Fiber length (Practically evaluated)

140

120 22.6, 113.19

100 20.1, 100.77 17.6, 88.38 80 15.1, 76.03

12.57, 63.73 60

FWHM in ps FWHM in 10.1, 51.5

40 7.54, 39.5 5, 27.95 20 2.51, 17.67 0, 12.5

0 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 Fiber Length in km

Figure 5.15: Full Width Half Maximum (FWHM) vs Fiber Length (Theoretically evaluated)

The above Figure 5.14 and Figure 5.15 shows the Full Width Half Maximum (FWHM) on the vertical axis and Fiber Length on the horizontal axis. The FWHM for each experimental fiber length signifies the amount of pulse broadening caused by the dispersion. The test values for fiber lengths are chosen from 퐿퐷 to 10 퐿퐷 where 퐿퐷 = 2.5135 km. The corresponding FWHM values are calculated practically shown in Figure 5.14 and the theoretical values are calculated using Equation 5.2 and are shown in Figure 5.15. These two graphs can be comparable as both graphs show elevated curves for increasing fiber lengths. The above FWHM values shows an increasing value which means distortion elevates as fiber length increases and worsens for a fiber length greater than 50 km as shown in Figure 5.13.

5.2.2 Self Phase Modulation Analysis: The self-phase modulation in the nonlinearities tab of the optical fiber properties as shown in Figure 5.3 is checked and the group velocity dispersion in the dispersion tab as shown in Figure 5.4 is unchecked.

As the GVD coefficient 훽2 is zero for this consideration, the nonlinear length 퐿푁퐿 can be calculated as, 1 퐿푁퐿 = (5.3) 훾푃0 The nonlinear coefficient, 훾 = 1.317 푊−1 ∕ 푘푚 .

Like the dispersion length 퐿퐷 calculated for the group velocity dispersion, the nonlinear length 퐿푁퐿 serves as convenient measure of nonlinear effects. If the distance of fiber is lesser than 퐿푁퐿, the Self Phase Modulation effects can be neglected. The power value is calculated by using the formula, 2 |훽| 2 푃 = 푁 2 = 0.30208 푁 [W] (5.4) 훾푇0

Figure 5.16: Effects of SPM for a fiber length of 15 km.

Figure 5.17: Effects of SPM for a fiber length of 30 km.

Figure 5.18: Effects of SPM for a fiber length of 45 km.

Figure 5.19: Effects of SPM for a fiber length of 60 km.

The nonlinearities are dominant at greater distances hence four arbitrary test fiber lengths 15 km, 30 km, 45 km and 60 km are chosen as shown above. The self-phase modulation creates more chirp in the frequency domain as the pulse travels for larger distances while the pulse remains undisturbed in the time domain. The effects are significant for fiber length greater than 퐿푁퐿 = 2.5135 km calculated using Equation 5.4 where 푃0 = 0.302 W. The phase ∅ associated with the pulse is shifted with the intensity I of the pulse as it travels along the fiber as shown below,

∅ = 2π (n − n I)L (5.5) λ NL eff

−α Where L = 1−e is known as the effective length, n is the linear index of refraction and eff α nNL is the nonlinear refractive index.

5.2.3 BER Analysis: The Bit Error Rate (BER) is considered a crucial parameter in defining the performance of any communication system. It is defined simply as the number of bits which are received at the output with an error. Most communication system serving high speed links require a minimum BER of 1*10-9.

Figure 5.20: min-BER (vertical axis) vs fiber length (horizontal axis) when GVD is acting alone.

Figure 5.20 shows the graph considering only the GVD factor, as seen from the graph as distance increases the BER is increased significantly. As fiber length increases the GVD has a significant effect on the error rate as the pulse broadens and becomes worse for a distance greater than 50 km.

Figure 5.21: min-BER (vertical axis) vs fiber length (horizontal axis) when SPM is acting alone. Figure 5.21 shows the graph when SPM is acting, apparently the Self Phase Modulation does not have any effect on BER. The SPM does not have any effect bin the temporal domain hence the bit error rate is not affected. As the fiber length increases the BER shows an increase in values as shown clearly in Figure 5.20. This shows that as the pulse travels along the fiber there are different effects in the form of GVD and SPM which can depreciate the output of a pulse resulting in more errors to creep up.

5.2.4 GVD and SPM combined effect: As discussed in the previous chapters, the propagation of soliton requires both the group velocity dispersion and self-phase modulation. The interplay between these two parameters becomes a crucial discussion for the propagation of soliton. To simulate the soliton propagation, the PMD and the GVD tabs in the optical fiber properties are selected. The Birefringence type is changed to deterministic and the differential group delay is maintained 0 ps/km. The soliton propagation is observed for a fiber length between 0 to 30 km. The samples of the time domain visualizer are shown below for fiber lengths of 5 km, 15 km and 30 km.

Figure 5.22: Soliton pulse in time domain at 5 km.

Figure 5.23: Soliton pulse in time domain at 15 km.

Figure 5.24: Soliton pulse in time domain at 30 km. The pulse shape remains the same throughout the fiber and remains undistorted as observed at the end of the fiber.

20 18 18.2 18.1 18.1 18.14 18.17 18.2 18.24 18.3 18.32

5 Full Width Half Maximum (FWHM) in ps in (FWHM) MaximumHalf Width Full

0 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 Fiber Length in km

Figure 5.25: Fiber length vs Full Width Half Maximum when both SPM and GVD are acting.

The pulse width (vertical axis) versus the fiber length (horizontal axis) is shown in Figure 5.25 when both Self Phase Modulation and Group Velocity Dispersion are acting inside the fiber. When compared with Figure 5.14 and Figure 5.15, this graph does not show significant pulse broadening and the pulse almost stays the same throughout the fiber length as supported by theory of soliton.

Conclusion: This paper demonstrates the effects of GVD and SPM on an optical pulse inside an optical fiber. The Group Velocity Dispersion broadens the pulse in the time domain. The Self Phase Modulation does not have any effect in the time domain but changes the phase of a pulse in the frequency domain. The interplay between SPM and GVD results in the formation of a soliton which does not change their shape and amplitude and hence can be used for long haul communication. Hence soliton propagation shows huge potential for undistorted communication providing desired power levels.

References

[1] A survey on Dispersion Management in Optical Solitons in Optical Communication Systems: Revathy Nagesh, Rajesh Mohan R, Asha R S.

[2] John Senior, Optical fiber communication Principles and Practice, Third Edition.

[3] Govind P.Agarwal, Fiber-Optic Communication Systems, Third Edition, Wiley Interscience.

[4] S.Novikov, S.V.Manakov, L.P.Pilaevskii and V.E.Zakharov, Theory of Solitons: The Inverse Scattering method, First Edition, Springer-Verlag New York, LLC.

[5] Yuri S. Kivshar, Govind P.Agarwal, Optical Solitons From Fibers to Photonic Crystals, Academic Press. [6] Fundamental and Higher Order Solitons, Optiwave Optisystem Archives.