Chapter 7: Laser Amplification
==============================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 7.
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 7.2: Wave Propagation In An Atomic Medium
-------------------------------------------------

Problem 7.2.1 — Lineshapes for absorption and phase shift in ruby
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.2.2 — The V-E = 0 approximation in deriving the wave equation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.2.3 — Extending the Taylor approximation to higher-order terms in high-loss materials
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Section 7.3: The Paraxial Wave Equation
---------------------------------------

Problem 7.3.1 — Applying the paraxial-wave approximation to Gaussian beam propagation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Section 7.4: Single-Pass Laser Amplification
--------------------------------------------

Problem 7.4.1 — Amplification bandwidth for a Gaussian 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.

Problem 7.4.2 — Absorption linewidth for an absorbing 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.

Problem 7.4.3 — An alternative bandwidth definition for low-gain amplifiers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.4.4 — Testing for a Gaussian atomic lineshape
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.4.5 — Gain versus frequency for a cascaded amplifier plus 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 7.4.6 — Continuation of the previous problem: general evaluation ofpassband broadening in a laser amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.4.7 — Continuation of the previous problem: relationship between midband gain and phase shift derivatives?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.4.8 — Linewidth modulation spectroscopyv: a new experimental technique
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Section 7.5: Stimulated-Transition Cross Sections
-------------------------------------------------

Problem 7.5.1 — Practical expression for atomic oscillator strength
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.5.2 — Design considerations for a high-energy-storage laser medium
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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 7.5.3 — Measuring an inverted laser transition cross section
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.5.4 — Energy storage in a Nd.YAG rod
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.5.5 — Gain through a thin atomic layer near a mirror
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.

Section 7.6: Saturation Intensities In Laser Materials
------------------------------------------------------

Problem 7.6.1 — Saturation intensity for a three-level atomic 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 7.6.2 — Saturation lineshape for the reactive part of a homogeneous two-level atomic transition
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.6.3 — Saturation behavior in a two-level absorber including excited-state absorption
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.6.4 — Power balance versus intensity in a two-level saturable absorber
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7: Homogeneous Saturation In Laser Amplifiers
-------------------------------------------------------

Problem 7.7.1 — Input-output intensity curves for a saturable amplifier offline center
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.2 — Input versus output intensities for a saturable atomic absorber
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.3 — Signal penetration and saturation depth versus signal intensity in a homogeneous saturable absorber
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.4 — Obtaining an amplifier output intensity just equal to the available intensity of the laser medium
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.5 — Power output stabilization factor in a partially saturated laser amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.6 — Cross-saturation of a transversely double-pass laser amplifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.7 — Amplifier input-output curves for other forms of laser saturation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.8 — Signal transmission through two intermingled saturable absorber transitions
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 7.7.9 — Saturation effects on transverse beam profiles
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.