Chapter 6: Laser Pumping and Population Inversion
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Source: Anthony E. Siegman, *Lasers* (1986), Chapter 6.
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 6.1: Steady-State Laser Pumping And Population Inversion
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Problem 6.1.1 — Three-level system with two pumping signals applied
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.1.2 — Population inversion versus pumping in an uupper-leveP three-level laser system
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.1.3 — Cascade pumping of a four-level laser 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 6.1.4 — Analysis of a five-level laser 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 6.1.5 — Laser refrigeration?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.2: Laser Gain Saturation
----------------------------------

Problem 6.2.1 — Transient response in the simplified laser-pumping model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.2.2 — Laser inversion and saturation including degeneracies
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.2.3 — Optically pumped laser absorber
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.2.4 — Simultaneous pumping into both the upper and the lower laser levels
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.2.5 — Signal saturation behavior in the ideal three-level laser 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.

Section 6.3: Transient Laser Pumping
------------------------------------

Problem 6.3.1 — Transient pumping with a Gaussian time-varying pump pulse
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.3.2 — Ditto with an exponentially varying pump pulse
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 6.3.3 — Peak population inversion versus normalized pump pulsewidth
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.