Chapter 25: Laser Spiking and Mode Competition
==============================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 25.
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 25.1: Laser Spiking And Relaxation Oscillations
-------------------------------------------------------

Problem 25.1.1 — Phase plane description of spiking
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.1.2 — Dimensionless form for the spiking 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 25.1.3 — Spiking analysis for the ruby laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.1.4 — Step response of a "spiky" laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.1.5 — Controlling spiking by an external feedback loop?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.1.6 — Proof that spiking will always die out in a simple laser system
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.1.7 — Extended spiking analysis for a semiconductor laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.2: Laser Amplitude Modulation
----------------------------------------

Problem 25.2.1 — Linearized small-signal response to laser cavity loss modulation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.2.2 — Linearized small-signal response of the laser population difference
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.3: Laser Frequency Modulation And Frequency Switching
----------------------------------------------------------------

Problem 25.3.1 — Verifying the step-function frequency-shift analysis
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.3.2 — Frequency shift analysis for a nonideal phase ramp
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.4: Laser Mode Competition
------------------------------------

Problem 25.4.1 — Mode competition with partial spatial overlap
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 25.4.2 — Mode competition analysis including coupling or scattering between modes (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.
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 25.4.3 — Gain saturation with time delay (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.
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.