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
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.
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 \(\delta=2\pi m\).
3. Verification.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.
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. 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
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.
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 \(\delta=2\pi m\).
3. Verification.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.
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. 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
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.
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 \(\delta=2\pi m\).
3. Verification.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.
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. 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
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.
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 \(\delta=2\pi m\).
3. Verification.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.
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. 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
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.
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 \(\delta=2\pi m\).
3. Verification.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.
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. 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
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.
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 \(\delta=2\pi m\).
3. Verification.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.
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. 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
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.
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.
3. Verification.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.
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. 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
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.
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.
3. Verification.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.
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. 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
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.
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.
3. Verification.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.
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. 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
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.
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.
3. Verification.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.
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. 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
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.
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.
3. Verification.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.
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. 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
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.
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.
3. Verification.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.
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. 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
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.
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.
3. Verification.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.
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. 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
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.
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.
3. Verification.
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.
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. 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
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.
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.
3. Verification.
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.
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. 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
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.
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.
3. Verification.
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.
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. 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
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.
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.
3. Verification.
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.
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. 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
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.
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.
3. Verification.
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.
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. 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
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.
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.
3. Verification.
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.
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. 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.