Chapter 4: Atomic Rate Equations
================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 4.
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 4.3: Blackbody Radiation And Radiative Relaxation
---------------------------------------------------------

Problem 4.3.1 — Thermal equilibration in a two-level atomic system: purely radiative case
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use :math:`N_2/N_1=(g_2/g_1)e^{-h\nu/kT}` together with Planck or Stefan--Boltzmann only after converting all temperatures to kelvins.
Check that the excited-state fraction stays between zero and one and approaches the correct high- and low-temperature limits.

Section 4.4: Nonradiative Relaxation
------------------------------------

Problem 4.4.1 — Thermal equilibration: radiative and nonradiative contributions
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Section 4.5: Two-Level Rate Equations And Saturation
----------------------------------------------------

Problem 4.5.1 — Signal-power absorption by a collection of atoms: Where does the absorbed power go?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Problem 4.5.2 — Effect of a sinusoidally modulated saturating signal: linearized analysis
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Problem 4.5.3 — Effects of a square-wave modulated saturating signal
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Section 4.6: Multilevel Rate Equations
--------------------------------------

Problem 4.6.1 — Saturation of the lower transition in a general three-level atomic system
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Problem 4.6.2 — Ditto for saturation of the upper transition in a three-level system
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Problem 4.6.3 — Saturation of the 1-3 transition in a three-level system: no optical approximation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Problem 4.6.4 — Ditto, using the optical approximation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Problem 4.6.5 — Rate-equation analysis of a thermally pumped 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.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.