Chapter 8: Nonlinear Optics

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 8. 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 8.1 — phase matching and three-wave mixing: 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. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.2 — phase matching and three-wave mixing: 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. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.3 — phase matching and three-wave mixing: 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. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.4 — phase matching and three-wave mixing: 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. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.5 — phase matching and three-wave mixing: 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. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.6 — phase matching and three-wave mixing: 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. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.7 — phase matching and three-wave mixing: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.8 — phase matching and three-wave mixing: discussion

Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Write \(\Delta k=k_p-k_s-k_i\) with the correct ordinary or extraordinary indices, expand it in angle, temperature, or frequency as requested, and integrate the coupled amplitudes over the crystal length. At exact phase match the conversion is maximal; Manley–Rowe photon-flux differences must remain constant.

Problem 8.9 — four-wave mixing and nonlinear susceptibility: 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. Select from \(P_i^{(3)}=\epsilon_0\chi_{ijkl}^{(3)}E_jE_kE_l\) only terms with the required frequency and wavevector. Insert them into the slowly varying wave equation, account for loss, and use isotropic-tensor symmetries before simplifying polarization cases. Frequency and momentum bookkeeping must close, the lossless coupled equations must conserve energy, and the nonlinear correction must vanish with field intensity.

Problem 8.10 — four-wave mixing and nonlinear susceptibility: 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. Select from \(P_i^{(3)}=\epsilon_0\chi_{ijkl}^{(3)}E_jE_kE_l\) only terms with the required frequency and wavevector. Insert them into the slowly varying wave equation, account for loss, and use isotropic-tensor symmetries before simplifying polarization cases. Frequency and momentum bookkeeping must close, the lossless coupled equations must conserve energy, and the nonlinear correction must vanish with field intensity.

Problem 8.11 — four-wave mixing and nonlinear susceptibility: 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. Select from \(P_i^{(3)}=\epsilon_0\chi_{ijkl}^{(3)}E_jE_kE_l\) only terms with the required frequency and wavevector. Insert them into the slowly varying wave equation, account for loss, and use isotropic-tensor symmetries before simplifying polarization cases. Frequency and momentum bookkeeping must close, the lossless coupled equations must conserve energy, and the nonlinear correction must vanish with field intensity.

Problem 8.12 — four-wave mixing and nonlinear susceptibility: 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. Select from \(P_i^{(3)}=\epsilon_0\chi_{ijkl}^{(3)}E_jE_kE_l\) only terms with the required frequency and wavevector. Insert them into the slowly varying wave equation, account for loss, and use isotropic-tensor symmetries before simplifying polarization cases. Frequency and momentum bookkeeping must close, the lossless coupled equations must conserve energy, and the nonlinear correction must vanish with field intensity.

Problem 8.13 — four-wave mixing and nonlinear susceptibility: 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. Select from \(P_i^{(3)}=\epsilon_0\chi_{ijkl}^{(3)}E_jE_kE_l\) only terms with the required frequency and wavevector. Insert them into the slowly varying wave equation, account for loss, and use isotropic-tensor symmetries before simplifying polarization cases. Frequency and momentum bookkeeping must close, the lossless coupled equations must conserve energy, and the nonlinear correction must vanish with field intensity.

Problem 8.14 — four-wave mixing and nonlinear susceptibility: 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. Select from \(P_i^{(3)}=\epsilon_0\chi_{ijkl}^{(3)}E_jE_kE_l\) only terms with the required frequency and wavevector. Insert them into the slowly varying wave equation, account for loss, and use isotropic-tensor symmetries before simplifying polarization cases. Frequency and momentum bookkeeping must close, the lossless coupled equations must conserve energy, and the nonlinear correction must vanish with field intensity.

Problem 8.15 — four-wave mixing and nonlinear susceptibility: 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. Select from \(P_i^{(3)}=\epsilon_0\chi_{ijkl}^{(3)}E_jE_kE_l\) only terms with the required frequency and wavevector. Insert them into the slowly varying wave equation, account for loss, and use isotropic-tensor symmetries before simplifying polarization cases. Frequency and momentum bookkeeping must close, the lossless coupled equations must conserve energy, and the nonlinear correction must vanish with field intensity.