Incoherently Pumped Fiber Optical Parametric Amplifiers 

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Chapter 2. Fundamental Concepts

Fiber Modes

Optical fibers are a type of circular dielectric waveguide, commonly made of silica glass.Typically, for a step-index fiber, it has a higher refractive index n1 core surrounding by a low refractive index n2 cladding. Fig. 2.1 shows the refractive index profile and cross section of a step-index fiber with a core radius of a. The guiding mechanism of the electromagnetic field in this type of optical fiber is known as total internal reflection.The eigenmodes are the solutions of electromagnetic field which propagates in a stepindex single mode fiber, and can be found by solving the wave equation with correct boundary conditions [73]. In circular waveguides, the eigenmodes are generally in either an HE or EH mode, depending on whether the electric field component or magnetic field component has a larger contribution in the propagation direction. The propagation constant (or wave number) β of HE and EH modes of a step-index optical fiber can be obtained by evaluating the following eigenvalue equations.

Chromatic Dispersion

In optics, chromatic dispersion is the phenomenon whereby the refractive index experienced by a propagating electromagnetic field in a medium is frequency-dependent. It plays a critical role in nonlinear optics. We will see later in this chapter how it significantly influences the interaction between optical fields of different frequencies. Since we are considering the electromagnetic field propagating solely in the fundamental spatial mode of an optical fiber, the dispersion only consists of two components: material dispersion and waveguide dispersion. The material dispersion originates from the frequency dependence of the linear susceptibility of the material, and it can be well described by the Sellmeier equation [74]; while, the waveguide dispersion can be understood by noting that the mode field diameter of the electromagnetic field in an optical fiber is frequency dependent, so the effective refractive index of the mode will change with frequency even in the absence of material dispersion [75]. To quantitatively account for the chromatic dispersion in optical fibers, it is customary to express the mode propagation constant β(ω) (or wave number) of the fundamental mode as a function of angular frequency ω.By expanding β(ω) into a Taylor series with respect to an arbitrary frequency ω0, we obtain the expression for the propagation constant in terms of dispersion parameters as follows.The first order derivative β1 is related to the Group Velocity vg = 1/β1 which describes the speed of an optical pulse propagating in an optical fiber. The second order derivative β2 is known as the Group Velocity Dispersion (GVD) parameter, and it is the parameter that describes how fast a pulse broadens when it propagates in an optical fiber. In the normal dispersion region where β2 > 0, optical pulses with longer center wavelengths travel faster than those with shorter center wavelengths; while the opposite occurs in the anomalous dispersion region where β2 < 0. Optical pulses experience a minimum group velocity dispersion at the Zero Dispersion Wavelength (ZDW) where β2 = 0. In fiber optics, the dispersion length is defined as where dΩ is the 3 dB spectral width of an optical pulse. It is commonly used to quantify the distance at which dispersion starts to have a significant effect on the optical pulse.

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Contents
Abstract
Acknowledgements 
Abbreviations
List of Publications
1 Introduction 
1.1 Background 
1.2 Recent Developments 
1.3 Objectives of Thesis 
1.4 Outline of Thesis
2 Fundamental Concepts 
2.1 Fiber Modes 
2.2 Chromatic Dispersion 
2.3 Fiber Nonlinearity
2.3.1 Kerr Effect
2.3.2 Self-Phase and Cross-Phase Modulation
2.3.3 Four-Wave Mixing
2.3.4 Stimulated Raman Scattering
2.4 Propagation Equations 
2.4.1 Nonlinear Schr ¨odinger Equation
2.4.2 Coupled-Mode Equations
2.5 Modulation Instability and the Raman Effect 
2.5.1 Undepleted Pump Linear Analysis
2.5.2 Effect of Pump Depletion
2.6 Summary 
3 Incoherently Pumped Fiber Optical Parametric Amplifiers 
3.1 Theory 
3.1.1 Temporally Incoherent Light
3.1.2 Gain Spectrum of an Incoherently Pumped Parametric Amplifier .
3.1.3 Gain Statistics of an Incoherently Pumped Parametric Amplifier
3.2 Experiment 
3.2.1 Temporally Incoherent Light Source
3.2.2 Experimental Setup for Gain Spectrum Measurement
3.2.3 Results of Experimental Gain Spectrum
3.2.4 Experimental Setup for Gain Statistics Measurement
3.2.5 Result of Experimental Gain Statistics .
3.3 Summary and Discussion
4 Cascaded Four-Wave Mixing
4.1 Cascaded Four-Wave Mixing 
4.1.1 Theory
4.1.2 Numerical Modeling
4.2 Observation of Cascaded Four-Wave Mixing 
4.2.1 Experimental Setup
4.2.2 Results
4.2.3 Dispersive Waves from a Pump in Normal Dispersion Regime
4.3 Cascaded Bragg Scattering
4.3.1 Theory
4.3.2 Numerical Modeling
4.3.3 Experimental Setup
4.3.4 Results
4.4 Summary and Discussion 
5 Fiber Optical Parametric Oscillators 
5.1 Theory 
5.1.1 Tunable Parametric Gain
5.1.2 Oscillator Cavity
5.1.3 Threshold and Conversion Efficiency
5.2 High Power Fiber Optical Parametric Oscillator 
5.2.1 Experimental Implementation
5.2.2 Results
5.3 High Conversion Fiber Optical Parametric Oscillator 
5.3.1 Numerical Modeling
5.3.2 Experimental Implementation
5.3.3 Results
5.4 Summary and Discussion 
6 Temporal Symmetry Breaking 
6.1 Fiber Ring Cavity 
6.2 Numerical Modeling 
6.2.1 Influence of Third Order Dispersion
6.3 Experimental Observation
6.3.1 Experimental Setup
6.3.2 Results
6.4 Summary and Discussion
7 Conclusion and Outlook
Bibliography 

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Novel χ (3) Phenomena in Optical Fibers and Fiber Cavities

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