Chapter 26: Laser Q-Switching
Source: Anthony E. Siegman, Lasers (1986), Chapter 26. 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 26.2: Active Q-Switching: Rate-Equation Analysis
Problem 26.2.1 — Calculating the Q-switched energy efficiency
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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Problem 26.2.2 — Two-step, two-pulse Q-switching
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 26.2.3 — Minimizing the Q-switched pulsewidth
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 26.2.4 — Q-switching performance versus tuning off line center
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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Problem 26.2.5 — Linearly opening slow Q-switch
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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Problem 26.2.6 — Sinusoidally opening slow Q-switch
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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Problem 26.2.7 — Double pulsing in a slowly opening Q-switched 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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Section 26.4: Repetitive Laser Q-Switching
Problem 26.4.1 — Plotting theoretical curves for repetitively switched lasers
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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Problem 26.4.2 — Transient start-up of a repetitively switched lasers
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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Problem 26.4.3 — Optimizing the cavity coupling in a repetitively switched 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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Problem 26.4.4 — Analytic result for repetition rate equals atomic decay rate
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.
During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss.
3. Verification.
Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted 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. During hold-off set the cavity photon number to zero and integrate inversion buildup; after the switch opens, integrate the coupled photon–inversion equations with the high-Q cavity loss. Confirm that pulse energy does not exceed stored inversion energy and that shortening the cavity lifetime raises peak power at fixed extracted energy.
Section 26.5: Mode Selection In Q-Switched Lasers
Problem 26.5.1 — Gain discrimination requirement for good mode discrimination
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.