Chapter 15: Semiconductor Lasers: Theory and Applications

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 15. 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 15.1 — semiconductor gain and modulation bandwidth: 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. Linearize carrier and photon rate equations about steady state, solve the resulting two-by-two frequency-domain system, and identify relaxation frequency and damping from the denominator poles. The dc limit must agree with the slope efficiency, and all stable small-signal poles must lie in the decaying half-plane.

Problem 15.2 — semiconductor gain and modulation bandwidth: 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. Linearize carrier and photon rate equations about steady state, solve the resulting two-by-two frequency-domain system, and identify relaxation frequency and damping from the denominator poles. The dc limit must agree with the slope efficiency, and all stable small-signal poles must lie in the decaying half-plane.

Problem 15.3 — semiconductor gain and modulation bandwidth: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Linearize carrier and photon rate equations about steady state, solve the resulting two-by-two frequency-domain system, and identify relaxation frequency and damping from the denominator poles. The dc limit must agree with the slope efficiency, and all stable small-signal poles must lie in the decaying half-plane.

Problem 15.4 — semiconductor gain and modulation bandwidth: 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. Linearize carrier and photon rate equations about steady state, solve the resulting two-by-two frequency-domain system, and identify relaxation frequency and damping from the denominator poles. The dc limit must agree with the slope efficiency, and all stable small-signal poles must lie in the decaying half-plane.

Problem 15.5 — laser spectra, chirp, threshold, and device design: plot

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. Evaluate material gain from the joint density of states and occupation difference, then combine it with confinement and mirror loss for threshold. For modulation, integrate instantaneous frequency to obtain phase and expand the AM–FM field into spectral sidebands. Net modal gain must equal total loss at threshold; carrier density, current, and sideband powers must be nonnegative, and total spectral power must match the time-domain average.

Problem 15.6 — laser spectra, chirp, threshold, and device design: 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. Evaluate material gain from the joint density of states and occupation difference, then combine it with confinement and mirror loss for threshold. For modulation, integrate instantaneous frequency to obtain phase and expand the AM–FM field into spectral sidebands. Net modal gain must equal total loss at threshold; carrier density, current, and sideband powers must be nonnegative, and total spectral power must match the time-domain average.

Problem 15.7 — laser spectra, chirp, threshold, and device design: 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. Evaluate material gain from the joint density of states and occupation difference, then combine it with confinement and mirror loss for threshold. For modulation, integrate instantaneous frequency to obtain phase and expand the AM–FM field into spectral sidebands. Net modal gain must equal total loss at threshold; carrier density, current, and sideband powers must be nonnegative, and total spectral power must match the time-domain average.

Problem 15.8 — laser spectra, chirp, threshold, and device design: 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. Evaluate material gain from the joint density of states and occupation difference, then combine it with confinement and mirror loss for threshold. For modulation, integrate instantaneous frequency to obtain phase and expand the AM–FM field into spectral sidebands. Net modal gain must equal total loss at threshold; carrier density, current, and sideband powers must be nonnegative, and total spectral power must match the time-domain average.

Problem 15.9 — laser spectra, chirp, threshold, and device design: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Evaluate material gain from the joint density of states and occupation difference, then combine it with confinement and mirror loss for threshold. For modulation, integrate instantaneous frequency to obtain phase and expand the AM–FM field into spectral sidebands. Net modal gain must equal total loss at threshold; carrier density, current, and sideband powers must be nonnegative, and total spectral power must match the time-domain average.

Problem 15.10 — laser spectra, chirp, threshold, and device design: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Evaluate material gain from the joint density of states and occupation difference, then combine it with confinement and mirror loss for threshold. For modulation, integrate instantaneous frequency to obtain phase and expand the AM–FM field into spectral sidebands. Net modal gain must equal total loss at threshold; carrier density, current, and sideband powers must be nonnegative, and total spectral power must match the time-domain average.

Problem 15.11 — laser spectra, chirp, threshold, and device design: plot

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. Evaluate material gain from the joint density of states and occupation difference, then combine it with confinement and mirror loss for threshold. For modulation, integrate instantaneous frequency to obtain phase and expand the AM–FM field into spectral sidebands. Net modal gain must equal total loss at threshold; carrier density, current, and sideband powers must be nonnegative, and total spectral power must match the time-domain average.