Chapter 13: Waveguide Coupling
Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 13. 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 13.1 — waveguide perturbation and modal orthogonality: 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. Project the perturbed Maxwell operator onto normalized unperturbed modes. The first-order propagation shift is an overlap integral of \(\Delta n^2|E|^2\); integration by parts and boundary decay establish orthogonality and power additivity. The shift must vanish with the perturbation, preserve units of inverse length, and give no cross power between orthogonal lossless modes.
Problem 13.2 — waveguide perturbation and modal orthogonality: 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. Project the perturbed Maxwell operator onto normalized unperturbed modes. The first-order propagation shift is an overlap integral of \(\Delta n^2|E|^2\); integration by parts and boundary decay establish orthogonality and power additivity. The shift must vanish with the perturbation, preserve units of inverse length, and give no cross power between orthogonal lossless modes.
Problem 13.3 — waveguide perturbation and modal orthogonality: 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. Project the perturbed Maxwell operator onto normalized unperturbed modes. The first-order propagation shift is an overlap integral of \(\Delta n^2|E|^2\); integration by parts and boundary decay establish orthogonality and power additivity. The shift must vanish with the perturbation, preserve units of inverse length, and give no cross power between orthogonal lossless modes.
Problem 13.4 — waveguide perturbation and modal orthogonality: 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. Project the perturbed Maxwell operator onto normalized unperturbed modes. The first-order propagation shift is an overlap integral of \(\Delta n^2|E|^2\); integration by parts and boundary decay establish orthogonality and power additivity. The shift must vanish with the perturbation, preserve units of inverse length, and give no cross power between orthogonal lossless modes.
Problem 13.5 — waveguide perturbation and modal orthogonality: 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. Project the perturbed Maxwell operator onto normalized unperturbed modes. The first-order propagation shift is an overlap integral of \(\Delta n^2|E|^2\); integration by parts and boundary decay establish orthogonality and power additivity. The shift must vanish with the perturbation, preserve units of inverse length, and give no cross power between orthogonal lossless modes.
Problem 13.6 — waveguide perturbation and modal orthogonality: 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. Project the perturbed Maxwell operator onto normalized unperturbed modes. The first-order propagation shift is an overlap integral of \(\Delta n^2|E|^2\); integration by parts and boundary decay establish orthogonality and power additivity. The shift must vanish with the perturbation, preserve units of inverse length, and give no cross power between orthogonal lossless modes.
Problem 13.7 — waveguide perturbation and modal orthogonality: 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. Project the perturbed Maxwell operator onto normalized unperturbed modes. The first-order propagation shift is an overlap integral of \(\Delta n^2|E|^2\); integration by parts and boundary decay establish orthogonality and power additivity. The shift must vanish with the perturbation, preserve units of inverse length, and give no cross power between orthogonal lossless modes.
Problem 13.8 — coupled modes, supermodes, and arrays: 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 \(d\mathbf A/dz=-iH\mathbf A\), include phase mismatch on the diagonal and overlap coupling off diagonal, then diagonalize \(H\). Propagate its eigenmodes and coherently sum array fields for the far pattern. A Hermitian lossless coupling matrix must conserve total modal power; identical two-guide eigenvalues must split symmetrically about the uncoupled propagation constant.
Problem 13.9 — coupled modes, supermodes, and arrays: 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 \(d\mathbf A/dz=-iH\mathbf A\), include phase mismatch on the diagonal and overlap coupling off diagonal, then diagonalize \(H\). Propagate its eigenmodes and coherently sum array fields for the far pattern. A Hermitian lossless coupling matrix must conserve total modal power; identical two-guide eigenvalues must split symmetrically about the uncoupled propagation constant.
Problem 13.10 — coupled modes, supermodes, and arrays: 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 \(d\mathbf A/dz=-iH\mathbf A\), include phase mismatch on the diagonal and overlap coupling off diagonal, then diagonalize \(H\). Propagate its eigenmodes and coherently sum array fields for the far pattern. A Hermitian lossless coupling matrix must conserve total modal power; identical two-guide eigenvalues must split symmetrically about the uncoupled propagation constant.
Problem 13.11 — coupled modes, supermodes, and arrays: discussion
Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Write \(d\mathbf A/dz=-iH\mathbf A\), include phase mismatch on the diagonal and overlap coupling off diagonal, then diagonalize \(H\). Propagate its eigenmodes and coherently sum array fields for the far pattern. A Hermitian lossless coupling matrix must conserve total modal power; identical two-guide eigenvalues must split symmetrically about the uncoupled propagation constant.
Problem 13.12 — coupled modes, supermodes, and arrays: 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 \(d\mathbf A/dz=-iH\mathbf A\), include phase mismatch on the diagonal and overlap coupling off diagonal, then diagonalize \(H\). Propagate its eigenmodes and coherently sum array fields for the far pattern. A Hermitian lossless coupling matrix must conserve total modal power; identical two-guide eigenvalues must split symmetrically about the uncoupled propagation constant.
Problem 13.13 — coupled modes, supermodes, and arrays: 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 \(d\mathbf A/dz=-iH\mathbf A\), include phase mismatch on the diagonal and overlap coupling off diagonal, then diagonalize \(H\). Propagate its eigenmodes and coherently sum array fields for the far pattern. A Hermitian lossless coupling matrix must conserve total modal power; identical two-guide eigenvalues must split symmetrically about the uncoupled propagation constant.