Chapter 20: Complex Paraxial Wave Optics
========================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 20.
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 20.2: Gaussian Beams And Abcd Matrices
----------------------------------------------

Problem 20.2.1 — Bilinear transforms for Gaussian beams in ABCD matrix systems
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 20.2.2 — Mirror design specification for a small He-Ne laser—ABCD analysis
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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 20.2.3 — Tolerances on the mirror design specification
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
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 20.2.4 — Focusing into a dielectric sample—ABCD analysis
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 20.4: Complex Paraxial Optics
-------------------------------------

Problem 20.4.1 — Square root of an ABCD matrix
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 20.5: Complex Hermite-Gaussian Modes
--------------------------------------------

Problem 20.5.1 — Verifying the cascading properties of complex ABCD matrices
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
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 20.5.2 — Complex Gaussian eigenmode of a complex Gaussian duct
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 20.5.3 — Biorthogonal modes in Gaussian ducts
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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 20.5.4 — Initial excitation of a complex Gaussian duct
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.