Chapter 12: Fundamentals of Laser Oscillation
=============================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 12.
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 12.1: Oscillation Threshold Conditions
----------------------------------------------

Problem 12.1.1 — Off-resonance regenerative amplification through an oscillating laser?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Section 12.2: Oscillation Frequency And Frequency Pulling
---------------------------------------------------------

Problem 12.2.1 — Number of modes in a doppler-broadened laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.2.2 — Laser cavity design for at least one and not more than three simultaneous axial modes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.2.3 — Length considerations for He-Ne laser design
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.2.4 — Mode pulling of the axial-mode spacing in different types of lasers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Section 12.3: Laser Output Power
--------------------------------

Problem 12.3.1 — Optimum coupling analysis for a unidirectional ring-cavity laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.3.2 — Internal losses and optimum coupling in real lasers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.3.3 — Laser power output versus tuning
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.3.4 — Laser oscillator with both saturable gain and saturable loss
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Problem 12.3.5 — Cross coupling between oscillation power and a separately injected signal
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.3.6 — Second-harmonic output coupling
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Section 12.4: Large Output Coupling Analysis
--------------------------------------------

Problem 12.4.1 — Optimum output coupling for a large-gain Rigrod-type laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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.

Problem 12.4.2 — Total power output from a high-gain laser oscillator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Problem 12.4.3 — Dual output coupling values for a high-gain laser oscillator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Problem 12.4.4 — Rigrod analysis of a one-way ring-laser oscillator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.4.5 — Two-segment ring-laser oscillator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :math:`G_{rt}=|G_{rt}|e^{j\Phi}`; resonance requires :math:`\Phi=2\pi q`, and threshold requires :math:`|G_{rt}|=1`.
Check the passive-cavity limit, energy conservation at every mirror, and that added loss raises rather than lowers threshold.

Problem 12.4.6 — Gain saturation in a high-gain, double-pass laser amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.