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
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.
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.
3. Verification.
The resonant gain–bandwidth tradeoff must narrow the peak as round-trip gain approaches threshold, while the passive limit recovers the etalon.
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. 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
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.
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.
3. Verification.
The resonant gain–bandwidth tradeoff must narrow the peak as round-trip gain approaches threshold, while the passive limit recovers the etalon.
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. 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
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.
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 \(\beta=R_m\tau_{sp}\).
3. Verification.
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.
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. 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
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.
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 \(\beta=R_m\tau_{sp}\).
3. Verification.
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.
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. 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
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.
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 \(\beta=R_m\tau_{sp}\).
3. Verification.
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.
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. 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 17.6 — amplified links and quantum-limited SNR: design
Brief solution
1. Method.
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits.
2. Decisive step.
Write the complete power split, amplifier saturation, and receiver noise budget before maximizing the integer branch product. For the quantum limit, include both zero-point sidebands that beat with the signal and average their independent phases.
3. Verification.
No branch may exceed the stated amplifier output, and the derived SNR degradation must be consistent with the standard quantum-limited amplifier noise figure.
Show detailed steps
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits. Write the complete power split, amplifier saturation, and receiver noise budget before maximizing the integer branch product. For the quantum limit, include both zero-point sidebands that beat with the signal and average their independent phases. No branch may exceed the stated amplifier output, and the derived SNR degradation must be consistent with the standard quantum-limited amplifier noise figure.
Problem 17.7 — amplified links and quantum-limited SNR: 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.
Write the complete power split, amplifier saturation, and receiver noise budget before maximizing the integer branch product. For the quantum limit, include both zero-point sidebands that beat with the signal and average their independent phases.
3. Verification.
No branch may exceed the stated amplifier output, and the derived SNR degradation must be consistent with the standard quantum-limited amplifier noise figure.
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. Write the complete power split, amplifier saturation, and receiver noise budget before maximizing the integer branch product. For the quantum limit, include both zero-point sidebands that beat with the signal and average their independent phases. No branch may exceed the stated amplifier output, and the derived SNR degradation must be consistent with the standard quantum-limited amplifier noise figure.