Chapter 19: Stable Two-Mirror Resonators
Source: Anthony E. Siegman, Lasers (1986), Chapter 19. Use each section/problem identifier with the book; the original prompts are not reproduced here. Each entry gives the governing model, the decisive solution route, and a physical verification.
Section 19.1: Stable Gaussian Resonator Modes
Problem 19.1.1 — Another graphical representation for resonator mode stability
Brief solution
1. Method.
Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 19.1.2 — Symmetric cavity with central thin lens
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 19.1.3 — Standing-wave cavity fields
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Section 19.2: Important Stable Resonator Types
Problem 19.2.1 — The “stop band” in near-confocal resonators
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\).
3. Verification.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\). Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Problem 19.2.2 — Mode matching from one stable resonator into another, (a) The output beam from a
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 19.2.3 — Spot size adjustments in a near-hemispherical resonator
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\).
3. Verification.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Build the round-trip ABCD matrix and solve \(q=(Aq+B)/(Cq+D)\); for real systems apply \(|A+D|<2\) before selecting the physical root with \(\operatorname{Im}(1/q)<0\). Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.
Section 19.3: Gaussian Transverse Mode Frequencies
Problem 19.3.1 — Relationship between spot size and transverse mode frequency
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 19.3.2 — Conditions for transverse mode degeneracy
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Section 19.5: Gaussian Resonator Mode Losses
Problem 19.5.1 — “Spillover losses” for a Gaussian resonator mode
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Problem 19.5.2 — Criterion for the “shoulder” in finite-diameter resonator loss curves
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end.
3. Verification.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Represent the beam by \(q^{-1}=R^{-1}-j\lambda/(\pi w^2)\) and propagate it with \(q_2=(Aq_1+B)/(Cq_1+D)\); separate real and imaginary parts only at the end. Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.