Chapter 21: Generalized Paraxial Resonator Theory
=================================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 21.
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 21.1: Complex Paraxial Resonator Analysis
-------------------------------------------------

Problem 21.1.1 — Resonator analysis using symmetrized matrices
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.3: Real And Geometrically Unstable Resonators
--------------------------------------------------------

Problem 21.3.1 — Gaussian-beam evolution in a geometrically unstable system
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.4: Complex Stable And Unstable Resonators
----------------------------------------------------

Problem 21.4.1 — Inserting a weak Gaussian aperture into an arbitrary ring resonator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.4.2 — Practical design of a large-mode-volume resonator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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 21.4.3 — Convergence to the final stable mode in a Gaussian-aperture-stabilized resonator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.4.4 — Same problem, but for a negative branch 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.

Problem 21.4.5 — Mode parameters in a Gaussian-aperture-stabilized two-mirror standing-wave resonator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.4.6 — Gaussian aperture stabilization in a plane-mirror resonator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.5: Other General Properties Of Paraxial Resonators
-------------------------------------------------------------

Problem 21.5.1 — Magnifying and demagnifying eigenwaves in an unsymmetric ring unstable 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 21.6: Multielement Stable Resonator Designs
---------------------------------------------------

Problem 21.6.1 — Longitudinal magnification of a telescope
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.6.2 — Contours of constant m in the stability diagram
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.6.3 — Perturbation-insensitive cavity design
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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 21.6.4 — Planar resonator with a single intracavity lens
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.6.5 — Mode profile in a stable multielement 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.

Problem 21.6.6 — Spot size stability against focusing at a different plane
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.6.7 — Characteristics of various types of ray systems
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 21.7: Orthogonality Properties Of Optical Resonator Modes
-----------------------------------------------------------------

Problem 21.7.1 — Demonstrating the transpose relationship for propagation in opposite directions through a multiaperture resonator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Insert the aperture field in the Huygens--Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds.
Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 21.7.2 — Symmetrizing the optical resonator eigenproblem
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.