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

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 13.1.2 — More complicated discrete buildup calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.1.3 — Cavity lifetime in a semiconductor diode 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 13.1.4 — Exact buildup solution for a typical 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.

Section 13.2: Derivation Of The Cavity Rate Equation
----------------------------------------------------

Problem 13.2.1 — Cavity photon number in a real 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 13.2.2 — Effective width of a Lorentzian transition
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.2.3 — Examples of the cavity-mode density formula
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.3: Coupled Cavity And Atomic Rate Equations
------------------------------------------------------

Problem 13.3.1 — Alternative derivation of the K coefficient
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :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 13.3.2 — Thermodynamic implications of the "extra photon."
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.3.3 — More cavity-mode thermodynamics
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.3.4 — Initial noise value for laser oscillation buildup (research problem)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :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 13.3.5 — Coupled rate-equation analysis of a transverse flow 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.

Section 13.4: The Laser Threshold Region
----------------------------------------

Problem 13.4.1 — Exact solution for the inverted population
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.4.2 — Cavity mode number in a He-Ne 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 13.4.3 — Alternative expression for output-power variation through threshold
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.4.4 — Threshold analysis with a partially bottlenecked lower level
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.4.5 — Threshold behavior in a two-mode laser (research problem)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :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 13.4.6 — Threshold behavior in a partially inhomogeneous two-mode laser (research problem)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :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 13.5: Multiple-Mirror Cavities And Etalon Effects
---------------------------------------------------------

Problem 13.5.1 — Three-mirror cavity frequency expression
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.5.2 — Energy distribution in a multimirror cavity
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.5.3 — Analysis of the Fox-Smith interferometer
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.7: Bistable Optical Systems
--------------------------------------

Problem 13.7.1 — Bistable 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 13.7.2 — Critical condition for absorptive bistability
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.7.3 — Absorptive bistability at large nonlinearity
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.7.4 — Bistable ring absorber cavity
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 13.7.5 — Inhomogeneous absorptive bistability Calculate and plot the input-output intensity relationship for a purely absorptive
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 :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.