Chapter 19: Stable Two-Mirror Resonators
========================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 19.
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 19.1: Stable Gaussian Resonator Modes
---------------------------------------------

Problem 19.1.1 — Another graphical representation for resonator mode stability
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 19.1.2 — Symmetric cavity with central thin lens
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 19.1.3 — Standing-wave cavity fields
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Section 19.2: Important Stable Resonator Types
----------------------------------------------

Problem 19.2.1 — The "stop band" in near-confocal resonators
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Build the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`; for real systems apply :math:`|A+D|<2` before selecting the physical root with :math:`\operatorname{Im}(1/q)<0`.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.

Problem 19.2.2 — Mode matching from one stable resonator into another, (a) The output beam from a
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 19.2.3 — Spot size adjustments in a near-hemispherical resonator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Build the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`; for real systems apply :math:`|A+D|<2` before selecting the physical root with :math:`\operatorname{Im}(1/q)<0`.
Back-propagate the eigenmode through one round trip to verify self-consistency and compare its aperture loss with the assumed stability regime.

Section 19.3: Gaussian Transverse Mode Frequencies
--------------------------------------------------

Problem 19.3.1 — Relationship between spot size and transverse mode frequency
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 19.3.2 — Conditions for transverse mode degeneracy
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Section 19.5: Gaussian Resonator Mode Losses
--------------------------------------------

Problem 19.5.1 — "Spillover losses" for a Gaussian resonator mode
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 19.5.2 — Criterion for the "shoulder" in finite-diameter resonator loss curves
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.