Chapter 1: An Introduction to Lasers
====================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 1.
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 1.1: What Is A Laser?
-----------------------------

Problem 1.1.1 — Diagramming the electromagnetic spectrum
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Put every landmark on one logarithmic coordinate using :math:`c=f\lambda`, :math:`\tilde\nu=1/\lambda`, and :math:`E=hf`; convert units before taking logarithms.
Cross-check each point by converting back from its plotted coordinate and verify that frequency increases as wavelength decreases.

Section 1.4: Laser Amplification
--------------------------------

Problem 1.4.1 — Numerical values for the Boltzmann ratio
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.
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 1.5: Laser Pumping And Population Inversion
---------------------------------------------------

Problem 1.5.1 — Slightly more complicated laser pumping system
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Convert photon energy with :math:`h\nu=hc/\lambda`, obtain particle number from the stated density and volume, and apply the relevant population or energy balance.
Check dimensional consistency, population bounds, and conservation of energy before interpreting the numerical scale.

Section 1.7: Laser Output-Beam Properties
-----------------------------------------

Problem 1.7.1 — Fraunhofer (far field) aperture diffraction patterns
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Start from :math:`I=P/A`; for a diffraction-limited aperture use :math:`\theta\simeq1.22\lambda/D` and propagate the far-field area before computing field strength from :math:`I=\tfrac12c\epsilon_0E_0^2`.
Verify power conservation through every aperture and confirm the expected inverse-square far-field irradiance.

Problem 1.7.2 — Huygens' integral and the on-axis intensity in the far field
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Start from :math:`I=P/A`; for a diffraction-limited aperture use :math:`\theta\simeq1.22\lambda/D` and propagate the far-field area before computing field strength from :math:`I=\tfrac12c\epsilon_0E_0^2`.
Verify power conservation through every aperture and confirm the expected inverse-square far-field irradiance.

Section 1.11: Additional Problems for Chapter 1
-----------------------------------------------

Problem 1.11.1 — Energy storage and Q-switching in a solid-state laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Convert photon energy with :math:`h\nu=hc/\lambda`, obtain particle number from the stated density and volume, and apply the relevant population or energy balance.
Check dimensional consistency, population bounds, and conservation of energy before interpreting the numerical scale.

Problem 1.11.2 — Optical intensity in a focused laser-beam spot
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Start from :math:`I=P/A`; for a diffraction-limited aperture use :math:`\theta\simeq1.22\lambda/D` and propagate the far-field area before computing field strength from :math:`I=\tfrac12c\epsilon_0E_0^2`.
Verify power conservation through every aperture and confirm the expected inverse-square far-field irradiance.

Problem 1.11.3 — Stimulated transition rate for molecules in a CO2 laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Convert photon energy with :math:`h\nu=hc/\lambda`, obtain particle number from the stated density and volume, and apply the relevant population or energy balance.
Check dimensional consistency, population bounds, and conservation of energy before interpreting the numerical scale.

Problem 1.11.4 — Stored energy and energy output in a TEA CO2 laser
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Convert photon energy with :math:`h\nu=hc/\lambda`, obtain particle number from the stated density and volume, and apply the relevant population or energy balance.
Check dimensional consistency, population bounds, and conservation of energy before interpreting the numerical scale.

Problem 1.11.5 — Heating effects due to focused laser beams
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
Start from :math:`I=P/A`; for a diffraction-limited aperture use :math:`\theta\simeq1.22\lambda/D` and propagate the far-field area before computing field strength from :math:`I=\tfrac12c\epsilon_0E_0^2`.
Verify power conservation through every aperture and confirm the expected inverse-square far-field irradiance.

Problem 1.11.6 — Laser fusion: laser design and fundamental economics
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
Convert photon energy with :math:`h\nu=hc/\lambda`, obtain particle number from the stated density and volume, and apply the relevant population or energy balance.
Check dimensional consistency, population bounds, and conservation of energy before interpreting the numerical scale.

Problem 1.11.7 — Thermal light sources versus coherent light sources
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
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.