Chapter 17: Optical Amplifiers

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 17. 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 17.1 — Fabry–Perot amplifier gain and ripple: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Insert the single-pass power gain in the Airy transmission, expand the denominator near resonance, and solve its half-power condition for bandwidth. Convert ratios to decibels only at the end. The resonant gain–bandwidth tradeoff must narrow the peak as round-trip gain approaches threshold, while the passive limit recovers the etalon.

Problem 17.2 — Fabry–Perot amplifier gain and ripple: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Insert the single-pass power gain in the Airy transmission, expand the denominator near resonance, and solve its half-power condition for bandwidth. Convert ratios to decibels only at the end. The resonant gain–bandwidth tradeoff must narrow the peak as round-trip gain approaches threshold, while the passive limit recovers the etalon.

Problem 17.3 — spontaneous emission and waveguide beta factor: 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. Normalize one traveling mode to a photon in length \(L\), obtain its density of states from \(dk/d\omega=n_g/c\), and apply Fermi’s golden rule. Spatially average the dipole–mode overlap to identify effective area and \(\beta=R_m\tau_{sp}\). All modal beta factors must lie between zero and one and their sum cannot exceed unity; enlarging effective mode area must reduce coupling to one mode.

Problem 17.4 — spontaneous emission and waveguide beta factor: 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. Normalize one traveling mode to a photon in length \(L\), obtain its density of states from \(dk/d\omega=n_g/c\), and apply Fermi’s golden rule. Spatially average the dipole–mode overlap to identify effective area and \(\beta=R_m\tau_{sp}\). All modal beta factors must lie between zero and one and their sum cannot exceed unity; enlarging effective mode area must reduce coupling to one mode.

Problem 17.5 — spontaneous emission and waveguide beta factor: 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. Normalize one traveling mode to a photon in length \(L\), obtain its density of states from \(dk/d\omega=n_g/c\), and apply Fermi’s golden rule. Spatially average the dipole–mode overlap to identify effective area and \(\beta=R_m\tau_{sp}\). All modal beta factors must lie between zero and one and their sum cannot exceed unity; enlarging effective mode area must reduce coupling to one mode.