Chapter 12: Wave Propagation in Periodic Media
Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 12. 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 12.1 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.2 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.3 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.4 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.5 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.6 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.7 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.8 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.9 — Bloch waves and multilayer transfer matrices: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity.
3. Verification.
Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Show detailed steps
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. Multiply one unit-cell transfer matrix, use its unit determinant, and solve \(\cos(K\Lambda)=\tfrac12\operatorname{tr}M\). Propagate \(M^N\) or its eigenvectors to obtain bands and finite-stack reflectivity. Pass bands require real Bloch wavevector; stop bands require exponential behavior, while power flux and the transfer determinant remain conserved in lossless layers.
Problem 12.10 — Bragg coupled modes and reflection waveguides: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling.
3. Verification.
The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.
Show detailed steps
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. Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling. The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.
Problem 12.11 — Bragg coupled modes and reflection waveguides: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling.
3. Verification.
The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.
Show detailed steps
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. Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling. The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.
Problem 12.12 — Bragg coupled modes and reflection waveguides: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling.
3. Verification.
The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.
Show detailed steps
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. Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling. The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.
Problem 12.13 — Bragg coupled modes and reflection waveguides: derivation
Brief solution
1. Method.
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.
2. Decisive step.
Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling.
3. Verification.
The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.
Show detailed steps
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. Remove the rapid phase with a rotating amplitude transformation, solve the resulting constant-coefficient two-mode system, and apply boundary conditions at both ends. Use the overlap integral for radiation or reflection-waveguide coupling. The appropriate co- or contradirectional power invariant must be constant, and zero coupling must restore independent forward and backward waves.