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

2 Casey H. Zhang

3 Abstract

4 In optics, a 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 , 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.

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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.

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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:

휕퐴(푧,푡) 훽 휕2퐴 51 푗 − 2 + 훾|퐴|2퐴 = 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.

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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.

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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:

훾푃 푇2 75 푁2 = 푝 0 [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):

푧 푡 −푗훾퐴2 82 퐴(푧, 푡) = 퐴 푠푒푐ℎ ( ) 푒 02 [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:

푡 86 푃(푡) = 푃 푠푒푐ℎ2 ( ) [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|훽 | 89 퐸 = 2 [6] 푝 |훾|휏

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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.

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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.

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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.

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140 References

141 Ablowitz, M. J., Ladik, J. F., 1976, Nonlinear differential–difference equations and Fourier analysis.

142 Journal of Mathematical , 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, , and

149 Philosophical Magazine and Journal of Science, 422-443.

150 Scott Russell, J., 1844, "Report on waves". Fourteenth meeting of the British Association for the Advancement

151 of Science.

152 Shabat, A., Zakharov, V., 1972, Exact theory of two-dimensional self-focusing and one-dimensional

153 self-modulation of waves in nonlinear media. Soviet Physics JETP, 34, 62-69.

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