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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Propagate irradiance with :math:`dI/dz=g(I)I`, using :math:`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.