Chapter 16: Advanced Semiconductor Lasers
Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 16. 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 16.1 — quantum-well confinement and scaling: 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.
Solve the piecewise one-dimensional Schrödinger equation and match wavefunction and mass-weighted derivative at interfaces. Combine the resulting confinement with mirror loss and differential efficiency when scaling cavity length.
3. Verification.
Bound-state energies must lie inside the well and approach the infinite-well values as the barrier grows.
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. Solve the piecewise one-dimensional Schrödinger equation and match wavefunction and mass-weighted derivative at interfaces. Combine the resulting confinement with mirror loss and differential efficiency when scaling cavity length. Bound-state energies must lie inside the well and approach the infinite-well values as the barrier grows.
Problem 16.2 — quantum-well confinement and scaling: 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.
Solve the piecewise one-dimensional Schrödinger equation and match wavefunction and mass-weighted derivative at interfaces. Combine the resulting confinement with mirror loss and differential efficiency when scaling cavity length.
3. Verification.
Bound-state energies must lie inside the well and approach the infinite-well values as the barrier grows.
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. Solve the piecewise one-dimensional Schrödinger equation and match wavefunction and mass-weighted derivative at interfaces. Combine the resulting confinement with mirror loss and differential efficiency when scaling cavity length. Bound-state energies must lie inside the well and approach the infinite-well values as the barrier grows.
Problem 16.3 — quantum-well confinement and scaling: 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.
Solve the piecewise one-dimensional Schrödinger equation and match wavefunction and mass-weighted derivative at interfaces. Combine the resulting confinement with mirror loss and differential efficiency when scaling cavity length.
3. Verification.
Bound-state energies must lie inside the well and approach the infinite-well values as the barrier grows.
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. Solve the piecewise one-dimensional Schrödinger equation and match wavefunction and mass-weighted derivative at interfaces. Combine the resulting confinement with mirror loss and differential efficiency when scaling cavity length. Bound-state energies must lie inside the well and approach the infinite-well values as the barrier grows.
Problem 16.4 — distributed-feedback coupling and oscillation phase: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\).
3. Verification.
With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
Show detailed steps
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\). With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
Problem 16.5 — distributed-feedback coupling and oscillation phase: plot
Brief solution
1. Method.
Derive a dimensionless plotting expression first, evaluate the limiting values and resonance or cutoff points, and then sample densely enough to resolve the narrowest feature.
2. Decisive step.
Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\).
3. Verification.
With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
Show detailed steps
Derive a dimensionless plotting expression first, evaluate the limiting values and resonance or cutoff points, and then sample densely enough to resolve the narrowest feature. Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\). With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
Problem 16.6 — distributed-feedback coupling and oscillation phase: 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.
Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\).
3. Verification.
With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
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. Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\). With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
Problem 16.7 — distributed-feedback coupling and oscillation phase: 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.
Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\).
3. Verification.
With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
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. Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\). With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
Problem 16.8 — distributed-feedback coupling and oscillation phase: 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.
Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\).
3. Verification.
With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.
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. Solve the forward/backward coupled-mode equations with detuning, coupling \(\kappa\), and gain or loss. Apply both facet conditions; oscillation requires both the magnitude condition and a round-trip phase equal to an integer multiple of \(2\pi\). With \(\kappa o0\) the distributed stop band must disappear, and passive reciprocal reflection must satisfy the expected symmetry.