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An Analytical, Numerical, and Experimental Study of Solitons in Optical Fiber

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An Analytical, Numerical, and Experimental Study of Solitons in Optical Fiber 1 An Analytical, Numerical, and Experimental Study of Solitons in Optical Fiber 2 Casey H. Zhang 3 Abstract 4 In optics, a soliton is an optical field that remains the same through propagation due to a 5 cancellation between nonlinear and dispersive effects in optical fiber. Under certain conditions, the 6 Kerr nonlinearity exactly cancels out the dispersion, and the shape of the pulse is preserved. In this 7 study, the research objective was to find the pulse energy required to theoretically achieve a soliton 8 in the Corning® SMF-28e® Optical Fiber. We first consider the fundamental hyperbolic secant 9 soliton solution of the Nonlinear Schrödinger Equation, and then we consider the pulse energy, the 10 temporal integral of the optical power of the pulse, and derive the numerical value of the exact pulse 11 energy necessary to create a soliton in the SMF-28e®. Then, this pulse energy is tested using 12 Fiberdesk V4.0, a program that simulates the propagation of light by solving the Nonlinear 13 Schrödinger Equation using a split-step Fourier transform. For a soliton with a full-width at half 14 maximum (FWHM) pulse duration of 1 푝푠, the pulse energy function yields a pulse energy of 15 50.9 푝퐽. 16 17 18 19 20 21 22 23 24 1 25 1. Introduction 26 Optical fiber is a flexible fiber usually made of high quality silica or plastic. It is most often used 27 as a waveguide to guide light between the ends of the fiber. Optical fibers are usually created with a 28 transparent core that is enclosed by a transparent cladding material with a lower index of refraction, 29 so that light is kept in the core by total internal reflection. The difference between the indices of 30 refraction and the incident angle of the source light allow little light to be lost when travelling 31 through the fiber. Optical fiber has many applications in telecommunications due to its high 32 bandwidth, lack of electromagnetic interference, and low attenuation. A standard telecommunication 33 optical fiber is the Corning® SMF-28e® (single-mode fiber) from Corning Incorporated®. 34 A soliton is a solitary wave that maintains its shape while propagating at a constant speed. 35 Solitons were first described in the context of water waves by John Scott Russell who observed a 36 solitary wave in Union Canal in Scotland. He called this phenomenon the “Wave of Translation” 37 (Russell 1844). Russell’s investigation and the discussion that followed were resolved by Diederik 38 Korteweg and Gustav de Vries (1895) when they derived a nonlinear equation, now called the 39 Korteweg-de Vries (KdV) Equation, to model waves on shallow water surfaces (Korteweg et al. 40 1895). Their paper includes solitary wave solutions. In 1973, Akira Hasegawa and Frederick Tappert 41 from AT&T Bell Labs were the first to suggest that solitons could exist in optical fiber (Hasegawa et 42 al. 1973). In optics, a temporal soliton is an optical field that remains the same through propagation 43 due to a cancellation between nonlinear and dispersive effects in a spatially confined medium, such 44 as an optical fiber. Generally, the shape of an optical pulse changes depending on the Kerr effect 45 and dispersion. However, under certain conditions, the Kerr nonlinearity exactly cancels out the 46 dispersion, and the shape of the pulse is preserved. Solitons are the solutions of nonlinear dispersive 47 partial differential equations that describe physical systems, i.e., the Korteweg-de Vries (KdV) 48 Equation, the Nonlinear Schrödinger Equation (NSE), etc. 2 49 If both dispersion and self-phase modulation (due to the Kerr effect) effects act on the 50 pulse, the field is described by the Nonlinear Schrödinger Equation: 2 휕퐴(푧,푡) 훽2 휕 퐴 2 51 푗 − + 훾|퐴| 퐴 = 0 [1] 휕푧 2 휕푡2 52 퐴 is a function that describes the field in terms of 푧, position, and 푡, time. 훾 is the self-phase 53 modulation coefficient (nonlinearity coefficient) in 푟푎푑/(푊 푚). 훽2 is the group velocity dispersion 54 (GVD) of the material in 푠2/푚. The Nonlinear Schrödinger Equation (NSE) is a nonlinear variation 55 of the Schrödinger Equation, and it governs pulse propagation through a nonlinear medium. The 56 NSE is an integrable model. 57 58 2. Statement of the Problem 59 My research objective was to find and understand the pulse energy required to theoretically 60 achieve a soliton in the specific optical fiber that I studied, which was the Corning® SMF-28e® 61 Optical Fiber. This was accomplished through finding the fundamental (푁 = 1) hyperbolic secant 62 (푠푒푐ℎ) soliton solution of the Nonlinear Schrödinger Equation, finding the pulse energy, and then 63 calculating the pulse energy required for the Corning® SMF-28e® Optical Fiber. After the pulse 64 energy was found, Fiberdesk V4.0 was used as a way of experimentally studying the theoretical pulse 65 energy required to create a soliton. Fiberdesk V4.0 is a simulation program that simulates the 66 propagation of light by solving the extended Nonlinear Schrödinger Equation using the split-step 67 Fourier transform method. 68 69 70 71 3 72 3. Analytical Derivation of Pulse Energy 73 The wave function, 퐴, is given an 푠푒푐ℎ function of 퐴 = 푁푠푒푐ℎ(푡). 푁 is a parameter that 74 summarizes the interaction between the dispersion and nonlinearity: 2 2 훾푃푝푇0 75 푁 = [2] 훽2 76 푃푝 is the peak power of the pulse, and 푇0 is the pulse width. If 푁 ≪ 1, the nonlinear term can be 77 neglected. If 푁 ≫ 1, the nonlinear term will be much more evident than the dispersion. When the 78 parameter 푁 = 1, a fundamental soliton is generated. In cases where 푁 > 1, a higher order soliton 79 can be generated from the input. 80 The fundamental (푁 = 1) soliton pulse has a shape of (assuming higher order dispersion is not 81 present): 2푧 푡 −푗훾퐴0 82 퐴(푧, 푡) = 퐴 푠푒푐ℎ ( ) 푒 2 [3] 0 휏 83 When the soliton is normalized such that 퐴(0,0) = 1: 푧 푡 −푗훾 84 퐴(푧, 푡) = 푠푒푐ℎ ( ) 푒 2 [4] 휏 85 The optical power of the pulse is: 2 푡 86 푃(푡) = 푃 푠푒푐ℎ ( ) [5] 푝 휏 87 The pulse energy 퐸푝, the integral of optical power over time, of a soliton depends inversely on the 88 soliton pulse duration 휏: 2|훽2| 89 퐸 = [6] 푝 |훾|휏 90 91 92 93 4 94 4. Numerical Derivation of Pulse Energy 95 The full-width at half maximum (FWHM) pulse duration is approximately 1.76 × 휏, and 1 푝푠 96 was used as the FWHM pulse duration in the Fiberdesk V4.0 simulation. The Corning® SMF-28e® 푟푎푑 97 at 1550 푛푚 has a nonlinearity coefficient of 1.43 × 10−3 and a 훽 of −20660 푓푠2/푚. Thus, 푊 푚 2 98 1 푝푠 solitons have a pulse energy of 50.9 푝퐽. The pulse energy for solitons with different durations 99 can be found by substituting a different value of 휏 into the pulse energy equation. For example, a 100 10 푝푠 soliton would have a pulse energy of 5.09 푝퐽, and a 100 푓푠 soliton would have a pulse 101 energy of 509 푝퐽. 102 103 5. Conclusion 104 By finding the pulse energy equation for a soliton, I was able to substitute the nonlinearity 105 coefficient and group velocity dispersion values with values specific to the Corning® SMF-28e® 106 Optical Fiber. After substituting these values into the equation, I was able to calculate the pulse 107 energy necessary to create a 1 푝푠 soliton. This pulse energy is 50.9 푝퐽. 108 Optical fibers can be used to transmit light and information over long distances. Optical fiber- 109 based systems have, in the most part, replaced radio transmitter systems for data transmission. They 110 are widely used for telecommunication. Optical fiber is used for Internet traffic, long high-speed 111 local area networks, cable television, etc. Any signal that carries information contains a range of 112 frequencies. The propagation speed of a wave depends on the frequency, so transmitted pulses tend 113 to break up due to dispersive spreading. When this dispersion is cancelled by the nonlinear effects to 114 create a soliton, pulses can be transmitted over long distances, which is necessary in 115 telecommunication. 5 116 Future research may include the understanding of what occurs when the pulse energy is slightly 117 deviated from the exact soliton solution. Future research may also include the analysis of higher 118 order solitons. Higher order (푁>1) solitons do not have a preserved shape. Rather, their shapes vary 119 periodically. Higher order solitons can also experience perturbations due to higher order dispersion, 120 so they are not as stable as fundamental solitons. 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 6 140 References 141 Ablowitz, M. J., Ladik, J. F., 1976, Nonlinear differential–difference equations and Fourier analysis. 142 Journal of Mathematical Physics, 17, 1011. 143 Ablowitz, M. J., Segur, H., 1992, Solitons, Nonlinear Evolution Equations and Inverse Scattering. 144 Cambridge University Press, 721-725. 145 Hasegawa, A., Tappert, F., 1973, Transmission of stationary nonlinear optical pulses in dispersive 146 dielectric fibers, Applied Physics Letters, 142-144. 147 Korteweg, D. J., de Vries, G., 1895, XLI On the change of form of long waves advancing in a 148 rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and 149 Dublin Philosophical Magazine and Journal of Science, 422-443.
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