Chapter 4: Optical Resonators

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 4. 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 4.1 — Fabry–Perot transmission and finesse: 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. Sum the round-trip geometric series. With phase \(\delta=2n\omega d\cos\theta/c\), reduce the intensity to an Airy denominator and obtain width or finesse by expanding about \(\delta=2\pi m\). At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.2 — Fabry–Perot transmission and finesse: 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. Sum the round-trip geometric series. With phase \(\delta=2n\omega d\cos\theta/c\), reduce the intensity to an Airy denominator and obtain width or finesse by expanding about \(\delta=2\pi m\). At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.3 — Fabry–Perot transmission and finesse: 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. Sum the round-trip geometric series. With phase \(\delta=2n\omega d\cos\theta/c\), reduce the intensity to an Airy denominator and obtain width or finesse by expanding about \(\delta=2\pi m\). At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.4 — Fabry–Perot transmission and finesse: 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. Sum the round-trip geometric series. With phase \(\delta=2n\omega d\cos\theta/c\), reduce the intensity to an Airy denominator and obtain width or finesse by expanding about \(\delta=2\pi m\). At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.5 — Fabry–Perot transmission and finesse: 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. Sum the round-trip geometric series. With phase \(\delta=2n\omega d\cos\theta/c\), reduce the intensity to an Airy denominator and obtain width or finesse by expanding about \(\delta=2\pi m\). At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.6 — Fabry–Perot transmission and finesse: 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. Sum the round-trip geometric series. With phase \(\delta=2n\omega d\cos\theta/c\), reduce the intensity to an Airy denominator and obtain width or finesse by expanding about \(\delta=2\pi m\). At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.7 — Gaussian resonator modes and coupling: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Form the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\). Use \(0<(A+D+2)/4<1\) or the equivalent \(g_1g_2\) test before extracting waist position, spot size, aperture power, or overlap. The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.8 — Gaussian resonator modes and coupling: 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. Form the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\). Use \(0<(A+D+2)/4<1\) or the equivalent \(g_1g_2\) test before extracting waist position, spot size, aperture power, or overlap. The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.9 — Gaussian resonator modes and coupling: 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. Form the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\). Use \(0<(A+D+2)/4<1\) or the equivalent \(g_1g_2\) test before extracting waist position, spot size, aperture power, or overlap. The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.10 — Gaussian resonator modes and coupling: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Form the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\). Use \(0<(A+D+2)/4<1\) or the equivalent \(g_1g_2\) test before extracting waist position, spot size, aperture power, or overlap. The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.11 — Gaussian resonator modes and coupling: 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. Form the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\). Use \(0<(A+D+2)/4<1\) or the equivalent \(g_1g_2\) test before extracting waist position, spot size, aperture power, or overlap. The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.12 — Gaussian resonator modes and coupling: 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. Form the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\). Use \(0<(A+D+2)/4<1\) or the equivalent \(g_1g_2\) test before extracting waist position, spot size, aperture power, or overlap. The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.13 — Gaussian resonator modes and coupling: 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. Form the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\). Use \(0<(A+D+2)/4<1\) or the equivalent \(g_1g_2\) test before extracting waist position, spot size, aperture power, or overlap. The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.14 — dispersive etalons, reciprocity, and ring filters: 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. Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay. Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order. Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.15 — dispersive etalons, reciprocity, and ring filters: 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. Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay. Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order. Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.16 — dispersive etalons, reciprocity, and ring filters: 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. Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay. Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order. Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.17 — dispersive etalons, reciprocity, and ring filters: 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. Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay. Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order. Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.18 — dispersive etalons, reciprocity, and ring filters: 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. Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay. Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order. Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.19 — dispersive etalons, reciprocity, and ring filters: 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. Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay. Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order. Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.