Chapter 13: Oscillation Dynamics and Oscillation Threshold
Source: Anthony E. Siegman, Lasers (1986), Chapter 13. 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 13.1: Laser Oscillation Buildup
Problem 13.1.1 — Discrete step behavior of laser oscillation buildup
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.1.2 — More complicated discrete buildup calculation
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.1.3 — Cavity lifetime in a semiconductor diode laser
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.1.4 — Exact buildup solution for a typical laser
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Section 13.2: Derivation Of The Cavity Rate Equation
Problem 13.2.1 — Cavity photon number in a real laser
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.2.2 — Effective width of a Lorentzian transition
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.2.3 — Examples of the cavity-mode density formula
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Section 13.3: Coupled Cavity And Atomic Rate Equations
Problem 13.3.1 — Alternative derivation of the K coefficient
Brief solution
1. Method.
Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
2. Decisive step.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.3.2 — Thermodynamic implications of the “extra photon.”
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.3.3 — More cavity-mode thermodynamics
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.3.4 — Initial noise value for laser oscillation buildup (research problem)
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
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. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.3.5 — Coupled rate-equation analysis of a transverse flow laser
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Section 13.4: The Laser Threshold Region
Problem 13.4.1 — Exact solution for the inverted population
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.4.2 — Cavity mode number in a He-Ne laser
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.4.3 — Alternative expression for output-power variation through threshold
Brief solution
1. Method.
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
2. Decisive step.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.4.4 — Threshold analysis with a partially bottlenecked lower level
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.4.5 — Threshold behavior in a two-mode laser (research problem)
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
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. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.4.6 — Threshold behavior in a partially inhomogeneous two-mode laser (research problem)
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
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. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Section 13.5: Multiple-Mirror Cavities And Etalon Effects
Problem 13.5.1 — Three-mirror cavity frequency expression
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.5.2 — Energy distribution in a multimirror cavity
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.5.3 — Analysis of the Fox-Smith interferometer
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Section 13.7: Bistable Optical Systems
Problem 13.7.1 — Bistable laser oscillator
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.7.2 — Critical condition for absorptive bistability
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.7.3 — Absorptive bistability at large nonlinearity
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.7.4 — Bistable ring absorber cavity
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
Problem 13.7.5 — Inhomogeneous absorptive bistability Calculate and plot the input-output intensity relationship for a purely absorptive
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.
Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\).
3. Verification.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.
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. Form the complex round-trip factor \(G_{rt}=|G_{rt}|e^{j\Phi}\); resonance requires \(\Phi=2\pi q\), and threshold requires \(|G_{rt}|=1\). Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.