Chapter 24: Laser Dynamics: The Laser Cavity Equations
======================================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 24.
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 24.1: Derivation Of The Laser Cavity Equations
------------------------------------------------------

Problem 24.1.1 — Cavity equations with a magnetic-dipole driving polarization
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.1.2 — Equivalent lumped circuit equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Section 24.2: External Signal Sources
-------------------------------------

Problem 24.2.1 — Power dissipation in an externally excited cavity
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.2.2 — Calculating regenerative cavity gain from the lumped equivalent circuit model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 24.2.3 — Alternative derivation of the external coupling term
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.2.4 — Another alternative derivation of the external coupling term
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Section 24.4: Alternative Formulations Of The Laser Equations
-------------------------------------------------------------

Problem 24.4.1 — Obtaining previous results using the SVEA equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.4.2 — Obtaining previous results using the phase-amplitude equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Problem 24.4.3 — Calculating atomic response from the phase-amplitude equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.

Section 24.5: Cavity And Atomic Rate Equations
----------------------------------------------

Problem 24.5.1 — Deriving the rate equations from the phase-amplitude equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use coupled photon--inversion equations, :math:`\dot n=(gN-\gamma_c)n+S` and :math:`\dot N=R_p-\gamma_2N-gNn`; solve the steady state before linearizing the Jacobian.
Check the threshold limit and require negative real parts for both small-signal eigenvalues when a stable operating point is claimed.