Chapter 11: Laser Mirrors and Regenerative Feedback
Source: Anthony E. Siegman, Lasers (1986), Chapter 11. 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 11.1: Laser Mirrors And Beam Splitters
Problem 11.1.1 — Scattering matrix for a general dielectric slab
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 11.1.2 — Changes in the scattering matrix for different reference planes
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 11.1.3 — Derivation of the necessary matrix element relationships for a lossless reciprocal two-port
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 11.1.4 — Scattering matrix for a transmission line junction
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 11.1.5 — Transmission-line junction with a lumped shunt capacitance
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 11.1.6 — Three-port and five-port optical scattering systems?
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 11.1.7 — Impossibility of a completely matched three-port network
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 11.1.8 — Conditions for an N-port equal-amplitude beam splitter
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 11.1.9 — Synthesizing an arbitrary complex optical two-port (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 11.3: Resonance Properties Of Passive Optical Cavities
Problem 11.3.1 — Design specifications for a transmission etalon
Brief solution
1. Method.
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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
Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound. 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 11.3.2 — Angle tuning of a transmission etalon
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 11.3.3 — Calculating cavity parameters from measured transmission-reflection 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.
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 11.3.4 — Reflection properties of uncoated dielectric etalon mirrors
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 11.3.5 — Linewidth of a power reflectivity dip
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 11.3.6 — Reflection phase angle versus frequency
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 11.3.7 — Field magnification inside a resonant 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.
Section 11.5: Optical-Cavity Mode Frequencies
Problem 11.5.1 — Axial-mode spectrum for an optical cavity with an internal dielectric section
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 11.5.2 — Axial-mode spectrum including dispersion
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.
Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening.
3. Verification.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Expand \(k(\omega)\) about the carrier, retain the requested orders, and use \(v_g=(dk/d\omega)^{-1}\) with \(k''\) controlling quadratic dispersive broadening. Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.
Problem 11.5.3 — Resonance properties of an equilateral triangular dielectric prism
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 11.5.4 — Mirror spacing in an optically pumped thin dye 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 11.6: Regenerative Laser Amplification
Problem 11.6.1 — Reflection gain of a regenerative laser amplifier
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.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 11.6.2 — Phase angle versus frequency for a regenerative laser cavity amplifier
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.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
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. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 11.6.3 — Enhanced feedback diagram
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.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Section 11.7: Approaching Threshold: The Highly Regenerative Limit
Problem 11.7.1 — Power transmission through a laser cavity halfway between axial modes
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 11.7.2 — Gain sensitivity of a regenerative laser amplifier
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.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 11.7.3 — Skirt selectivity of a regenerative laser amplifier
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.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 11.7.4 — Output versus input for a regenerative laser amplifier with saturable internal gain
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.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
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. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 11.7.5 — Regenerative gain peaks for off-line-center axial modes
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.
Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Problem 11.7.6 — Transient reflection from a resonant 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 11.7.7 — Approach to threshold in the Schawlow-Townes model
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.