Chapter 9: Linear Pulse Propagation
===================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 9.
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 9.1: Phase And Group Velocities
---------------------------------------

Problem 9.1.1 — Time-bandwidth products for various optical pulseshapes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`k''` controlling quadratic dispersive broadening.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.

Section 9.2: The Parabolic Equation
-----------------------------------

Problem 9.2.1 — Parabolic equation derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`k''` controlling quadratic dispersive broadening.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.

Section 9.3: Group Velocity Dispersion And Pulse Compression
------------------------------------------------------------

Problem 9.3.1 — Phase shift versus frequency analysis for the Gires-Tournois interferometer
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`k''` controlling quadratic dispersive broadening.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.

Problem 9.3.2 — Usefulness of the Gires-Tournois interferometer?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`k''` controlling quadratic dispersive broadening.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.

Section 9.4: Phase And Group Velocities In Resonant Atomic Media
----------------------------------------------------------------

Problem 9.4.1 — Analysis of group-velocity slowing in the wings of a strong atomic transition
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`k''` controlling quadratic dispersive broadening.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.

Problem 9.4.2 — Phase and group velocity versus frequency in a mixed laser amplifier and atomic absorber medium
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 9.4.3 — Sensitivity of pulse compression to disperser length
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Expand :math:`k(\omega)` about the carrier, retain the requested orders, and use :math:`v_g=(dk/d\omega)^{-1}` with :math:`k''` controlling quadratic dispersive broadening.
Check the transform-limited and zero-dispersion limits, and conserve pulse energy when only phase is changed.

Section 9.5: Pulse Broadening And Gain Dispersion
-------------------------------------------------

Problem 9.5.1 — Pulse broadening on passing through a Fabry-Perot etalon
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 9.5.2 — Pulse propagation through mixed group-velocity dispersion and gain dispersion
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 9.5.3 — Pulse propagation and distortion tuned on the side of an amplifying atomic transition
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.