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  • Nonlinear Optical Phenomena in Multimode Fibers

    by

    Shaival V. Buch

    Submitted in Partial Ful�llment of the

    Requirements for the Degree

    Doctor of Philosophy

    Supervised by Professor Govind P. Agrawal

    The Institute of Optics

    Arts, Science and Engineering

    Edmund A. Hajim School of Engineering and Applied Sciences

    University of Rochester

    Rochester, New York

    2018

  • Table of Contents

    Biographical Sketch .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

    Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

    Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

    Contributors and Funding Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viii

    List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

    List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiii

    1. Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1 Historical perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problems studied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2. Theory of multimode propagation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2.1 Generalized multimode nonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Normalized MM-NLSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Numerically solving the coupled NLSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.3.1 Split-step Fourier method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Fourth-order Runge-Kutta in the Interaction picture . . . . . . . . . . . . . . . . . . 11

    3. Solitons in few-mode �bers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    3.1 Temporal stability of solitons in few-mode �bers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1 Single soliton propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Multiple soliton propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    3.2 Temporal soliton trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Spatiotemporal solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    4. Impact of Kerr nonlinearity on intermodal soliton interaction .. . . . . . . . . . 21

    4.1 Soliton interaction in �rst-order degenerate modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Impact of pulse parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    4.2.1 Impact of relative phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.2 Impact of temporal pulse separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.3 Impact of relative amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    4.3 Soliton interaction in nearly degenerate modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

  • TABLE OF CONTENTS iii

    5. Impact of Raman scattering on soliton interaction .. . . . . . . . . . . . . . . . . . . . . . . 32

    5.1 Review of Raman scattering and its modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Interaction of two in-phase solitons in SMFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    5.2.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2.2 Some analytical insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    5.3 Impact of relative phase and amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.3.1 Relative phase di�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.3.2 Relative amplitude di�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    5.4 Impact of Raman scattering on intermodal soliton interaction . . . . . . . . . . . . . . . . 43

    6. Spectral compression and broadening in MMFs .. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    6.1 Suppression of spectral broadening in normal dispersion few-mode �bers . . . . 47 6.1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    6.2 Experimental results of nonlinear spectral broadening . . . . . . . . . . . . . . . . . . . . . . . . . 51

    7. Nonlinear propagation equations with random linear mode coupling.. . . 55

    7.1 Model for RLMC .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.2 Transfer matrix for a multimode �ber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    7.2.1 Physical considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.2.2 Numerically computing the transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    7.3 Averaged nonlinear propagation equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    8. Impact of linear coupling on nonlinear propagation .. . . . . . . . . . . . . . . . . . . . . . 64

    8.1 Fourth-order moments of transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.2 Averaged equations in di�erent coupling regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 8.3 Full numerical simulation of an SDM system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    9. Characterization of linear coupling in coupled-core �bers . . . . . . . . . . . . . . . . 72

    9.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

    Bibliography.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    Appendix 86

    A. Deriving kth component of nonlinear term with random coupling in-

    cluded .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

    B. Deriving nonlinear coe�cients for the averaged Manakov equation .. . . . 90

  • Biographical Sketch

    The author was born in Gandhinagar in the state of Gujarat in India. After �nishing high school in Vadodara, he enrolled in the Engineering Physics program at the Indian Institute of Technology Guwahati in Assam and obtained his Bachelor of Technology degree in 2012. In the fall of 2012, he joined The Institute of Optics, University of Rochester to pursue a Master of Science degree in Optics, which was awarded to him in 2013. In the fall of 2013, he enrolled in the PhD program at The Institute of Optics and joined Dr. Govind Agrawal's Nonlinear Photonics Group. He has worked on numerical modeling of nonlinear phenomena in multimode optical �bers and collaborated with Nokia Bell Labs to study light propagation through multimode lightwave systems. He was involved in a two-year experimental project on space optical communications using laser beam ampli�cation as part of a collaboration between NASA and the University of Rochester. He also worked on free-space coherent optical communication systems as an intern at Mitsubishi Electric Research Labs.

    The following publications were a result of work conducted during doctoral

    study:

    S. Buch and G. P. Agrawal,�Soliton stability and trapping in multimode �bers,� Opt. Lett., vol. 40, pp. 225-228 (2015).

    S. Buch and G. P. Agrawal, �Intermodal soliton interaction in nearly degenerate modes of a multimode �ber,� J. Opt. Soc. Am. B, vol. 33, pp. 2217-2224 (2016).

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