Chapter 14: Nonlinear Optical Effects in Fibers

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 14. Use each problem number with the book; the original prompts are not reproduced. Each entry supplies the governing model, a decisive solution route, and an independent consistency check.

End-of-chapter problems

Problem 14.1 — self-phase modulation and dispersive propagation: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Use the nonlinear envelope equation \(\partial_zA+\beta_1\partial_tA+i\beta_2\partial_t^2A/2=i\gamma|A|^2A\). Separate nonlinear phase from dispersive transfer and transform Gaussian segments when the fiber is periodic. With \(\gamma=0\) pulse energy and the linear dispersion result must be recovered; with \(eta_2=0\) intensity stays fixed while phase accumulates.

Problem 14.2 — self-phase modulation and dispersive propagation: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Use the nonlinear envelope equation \(\partial_zA+\beta_1\partial_tA+i\beta_2\partial_t^2A/2=i\gamma|A|^2A\). Separate nonlinear phase from dispersive transfer and transform Gaussian segments when the fiber is periodic. With \(\gamma=0\) pulse energy and the linear dispersion result must be recovered; with \(eta_2=0\) intensity stays fixed while phase accumulates.

Problem 14.3 — self-phase modulation and dispersive propagation: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Use the nonlinear envelope equation \(\partial_zA+\beta_1\partial_tA+i\beta_2\partial_t^2A/2=i\gamma|A|^2A\). Separate nonlinear phase from dispersive transfer and transform Gaussian segments when the fiber is periodic. With \(\gamma=0\) pulse energy and the linear dispersion result must be recovered; with \(eta_2=0\) intensity stays fixed while phase accumulates.

Problem 14.4 — self-phase modulation and dispersive propagation: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Use the nonlinear envelope equation \(\partial_zA+\beta_1\partial_tA+i\beta_2\partial_t^2A/2=i\gamma|A|^2A\). Separate nonlinear phase from dispersive transfer and transform Gaussian segments when the fiber is periodic. With \(\gamma=0\) pulse energy and the linear dispersion result must be recovered; with \(eta_2=0\) intensity stays fixed while phase accumulates.

Problem 14.5 — self-phase modulation and dispersive propagation: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Use the nonlinear envelope equation \(\partial_zA+\beta_1\partial_tA+i\beta_2\partial_t^2A/2=i\gamma|A|^2A\). Separate nonlinear phase from dispersive transfer and transform Gaussian segments when the fiber is periodic. With \(\gamma=0\) pulse energy and the linear dispersion result must be recovered; with \(eta_2=0\) intensity stays fixed while phase accumulates.

Problem 14.6 — self-phase modulation and dispersive propagation: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Use the nonlinear envelope equation \(\partial_zA+\beta_1\partial_tA+i\beta_2\partial_t^2A/2=i\gamma|A|^2A\). Separate nonlinear phase from dispersive transfer and transform Gaussian segments when the fiber is periodic. With \(\gamma=0\) pulse energy and the linear dispersion result must be recovered; with \(eta_2=0\) intensity stays fixed while phase accumulates.

Problem 14.7 — soliton pulses, four-wave mixing, and polarization: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Insert the sech or Gaussian ansatz, evaluate its Fourier transform, and balance dispersive and nonlinear coefficients. For mixing, derive amplitudes from the resonant \(\chi^{(3)}\) polarization and impose the Manley–Rowe invariant. The transform widths must obey the appropriate time-bandwidth product, and lossless frequency conversion must conserve the weighted photon flux.

Problem 14.8 — soliton pulses, four-wave mixing, and polarization: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Insert the sech or Gaussian ansatz, evaluate its Fourier transform, and balance dispersive and nonlinear coefficients. For mixing, derive amplitudes from the resonant \(\chi^{(3)}\) polarization and impose the Manley–Rowe invariant. The transform widths must obey the appropriate time-bandwidth product, and lossless frequency conversion must conserve the weighted photon flux.

Problem 14.9 — soliton pulses, four-wave mixing, and polarization: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Insert the sech or Gaussian ansatz, evaluate its Fourier transform, and balance dispersive and nonlinear coefficients. For mixing, derive amplitudes from the resonant \(\chi^{(3)}\) polarization and impose the Manley–Rowe invariant. The transform widths must obey the appropriate time-bandwidth product, and lossless frequency conversion must conserve the weighted photon flux.

Problem 14.10 — soliton pulses, four-wave mixing, and polarization: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Insert the sech or Gaussian ansatz, evaluate its Fourier transform, and balance dispersive and nonlinear coefficients. For mixing, derive amplitudes from the resonant \(\chi^{(3)}\) polarization and impose the Manley–Rowe invariant. The transform widths must obey the appropriate time-bandwidth product, and lossless frequency conversion must conserve the weighted photon flux.

Problem 14.11 — soliton pulses, four-wave mixing, and polarization: derivation

Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Insert the sech or Gaussian ansatz, evaluate its Fourier transform, and balance dispersive and nonlinear coefficients. For mixing, derive amplitudes from the resonant \(\chi^{(3)}\) polarization and impose the Manley–Rowe invariant. The transform widths must obey the appropriate time-bandwidth product, and lossless frequency conversion must conserve the weighted photon flux.