Chapter 10: Nonlinear Optical Pulse Propagation
===============================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 10.
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 10.1: Pulse Amplification With Homogeneous Gain Saturation
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Problem 10.1.1 — Pulse input energy to saturate the pulse energy gain down to just half the initial unsaturated gain
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 10.1.2 — Leading-edge spike width for an infinitely sharp square input pulse
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 10.1.3 — Calculating input-output pulse profiles for Gaussian input pulses of varying pulse energy
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 10.1.4 — Pulse energy transmission through a saturable atomic absorber
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 10.1.5 — Measuring pulse saturation energies using photoacoustic spectroscopy
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 10.1.6 — Penetration depth versus energy for pulses traveling into a saturable absorber
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
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 10.1.7 — Pulse input-output and pulse energy extraction for a partially bottlenecked lower energy level (research
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.